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1ReliabilityModelingwithApplications9023_9789814571937_tp.indd15/11/135:36PM
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3ReliabilityModelingwithApplicationsEssaysinHonorofProfessorToshioNakagawaonHis70thBirthdayEditorsSyoujiNakamuraKinjoGakuinUniversity,JapanCunHuaQianNanjingUniversityofTechnology,ChinaMingchihChenFeJenCatholicUniversity,TaiwanWorldScientificNEWJERSEY•LONDON•SINGAPORE•BEIJING•SHANGHAI•HONGKONG•TAIPEI•CHENNAI9023_9789814571937_tp.indd25/11/135:36PM
4PublishedbyWorldScientificPublishingCo.Pte.Ltd.5TohTuckLink,Singapore596224USAoffice:27WarrenStreet,Suite401-402,Hackensack,NJ07601UKoffice:57SheltonStreet,CoventGarden,LondonWC2H9HEBritishLibraryCataloguing-in-PublicationDataAcataloguerecordforthisbookisavailablefromtheBritishLibrary.RELIABILITYMODELINGWITHAPPLICATIONSEssaysinHonorofProfessorToshioNakagawaonHis70thBirthdayCopyright©2014byWorldScientificPublishingCo.Pte.Ltd.Allrightsreserved.Thisbook,orpartsthereof,maynotbereproducedinanyformorbyanymeans,electronicormechanical,includingphotocopying,recordingoranyinformationstorageandretrievalsystemnowknownortobeinvented,withoutwrittenpermissionfromthepublisher.Forphotocopyingofmaterialinthisvolume,pleasepayacopyingfeethroughtheCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923,USA.Inthiscasepermissiontophotocopyisnotrequiredfromthepublisher.ISBN978-981-4571-93-7PrintedinSingapore
5September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookPrefaceProfessorToshioNakagawahasgreatlycontributedtothegrowthofava-rietyofMaintenancePoliciesinReliabilityTheoryduringfourdecades.Astheresults,hehaspublishedover200papersinresearchjournals,andsum-marizingthem,hasalreadypublished4booksandwillpublishsoononemorebookfromSpringerseriesinReliabilityEngineering,asshowninanimpressivePublicationListofChapter19.Furthermore,hehasorganizedaresearchgroupnamedNagoyaComputerandReliabilityResearch(NCRR)withtheprimeobjectiveofpresentingandwritingresearchpapersstud-iedbyeachmember.TheNCRRhascontinuedfor25yearsunexpectedlysince1989,andeachmemberhaspresentedactivelymanypapersunderhisleadershipatseveralinternationalworkshopsorganizedmainlybyPro-fessorShunjiOsaki,ProfessorHoangPham,ProfessorShigeruYamada,ProfessorTadashiDohiandotherdistinguishedresearchers.Inmemoryof20thanniversary,someresearchresultsofthemainmemberswerepub-lishedonthebookformonStochasticReliabilityModeling,OptimizationandApplicationsfromWorldScientificPublishingeditedwithProfessorSyoujiNakamurain2010.AftergraduatingNagoyaInstituteofTechnologyin1967,hejoinedDe-partmentofMathematics,MeijoUniversityinNagoya,gotaDoctorofEngineeringDegreefromKyotoUniversityin1977,andbecameaProfes-sorofDepartmentofIndustrialEngineering,AichiInstituteofTechnologyin1988.Heretiredhisteachingjobfor46yearsin2013andisnowaGuestProfessorofAichiInstituteofTechnology.However,hehopestocontinuetostudyreliabilitytheoryfromthistimeforth.Recently,recognizinghisacademicachievement,hereceivedServiceAwardfromInternationalSo-cietyofScienceandAppliedTechnologiesin2009,PresidentAwardfromAichiInstituteofTechnologyin2012,andReliabilityEngineeringAwardfromtheIEEEReliabilitySocietyJapanChapterin2012.v
6September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookviReliabilityModelingwithApplicationsIncelebrationofhis70thbirthday,wehavemadeaplanofpublishingthebooktitledonReliabilityModelingwithApplicationsfromWorldSci-entificPublishing,andcalledouttotheresearchersfamiliarwithhimatinternationalconferencesandworkshops.Fortunately,morethan30ex-cellentauthorsintheworldhavesupportedwillinglytothisplan,andthebookcanbepublishedbytheirpositivecooperation.Thebookiscomposedof4parts:MaintenancePolicieswith5papers,ReliabilityAnalysiswith4papers,ComputerSystemwith5papersandReliabilityApplicationswith4papers.Somepaperspresentnewinsightsorresultsforfuturestudies,givegoodoverviewsofcurrentresearches,andproposeusefultechniquesforpracticalapplicationsinreliabilityareas.Finally,ProfessorNakagawamakeshispublicationlistdividedinto8parts,andalso,giveshisfutureplanforfurtherstudiesinChapter19.Webelievethatthisbookcanserveasagoodtextbookandguidebookforstudents,engineersandresearchersinreliability.Wewouldliketoexpressoursincereappreciatetoallauthorstothisbook,andalso,Dr.XufengZhaoforhiskindhelpinwritingandtypingthisbook.Finally,wewouldliketothankforthesupportandEditorChelseaChin,WorldScientificPublishingforprovidingtheopportunitytopublishthisbook.SyoujiNakamura,KinjoGakuinUniversity,JapanCunHuaQian,NanjingUniversityofTechnology,ChinaMingchihChen,FeJenCatholicUniversity,Taiwan
7September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookContentsPrefacevMaintenancePolicies11.ADynamicProgrammingApproachforSequentialPreventiveMaintenancePolicieswithTwoFailureModes3HiroyukiOkamura,TadashiDohiandShunjiOsaki1Introduction..........................32SequentialImperfectPM...................53DPAlgorithm.........................74NumericalExamples.....................105ConcludingRemarks.....................152.SelectiveMaintenanceforComplexSystemsConsideringImperfectMaintenanceEfficiency17MayankPandey,YuLiu,MingJ.Zuo1Introduction..........................172SelectiveMaintenanceforBinarySystems.........212.1HazardAdjustmentFactor.............252.2SystemReliabilityEvaluation...........262.3SelectiveMaintenanceModeling..........272.4Results........................283SelectiveMaintenanceforMulti-stateSystems.......313.1SelectiveMaintenanceforMSSwithBinaryComponents.....................32vii
8September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookviiiReliabilityModelingwithApplications3.2SelectiveMaintenanceforMSSwithMultistateComponents.....................394Conclusion..........................463.RandomReplacementPolicies51WonYoungYunandLiLiu1Introduction..........................512RandomAgeReplacementPolicies.............522.1StandardAgeReplacementPolicy.........532.2RandomAgeReplacementPolicy.........542.3NumericalExamples................593PreventiveReplacementPolicieswithRandomOpportunityTimes......................613.1OpportunityAgeReplacementModel.......613.2Number-basedReplacementPolicy........623.3NumericalExamples................634ExtendedModels.......................645Conclusion..........................654.OptimalReplacementIntervalofaDualSystem67SatoshiMizutani1Introduction..........................672Modeling...........................683OptimalPolicies.......................723.1ReplacementinRandomFailurePeriod......723.2ReplacementinWearoutFailurePeriod......724NumericalExamples.....................744.1ReplacementinRandomFailurePeriod......754.2ReplacementinWearoutFailurePeriod......755Conclusions..........................765.CumulativeDamageModelswithRandomWorkingTimes79XufengZhao,CunhuaQianandShey-HueiSheu1Introduction..........................792ModelingandOptimization.................812.1WorkingNumberN................822.2ReplacementOverTimeTorLevelZ......83
9September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookContentsix2.3ReplacementFirstandLast............893Conclusions..........................96ReliabilityAnalysis996.ModulesofMulti-StateCoherentSystems—OrderTheoreticalRelations101FumioOhi1Introduction..........................1012Multi-stateCoherentSystems................1033Modules............................1074ConcludingRemarks.....................1147.CalculationAlgorithmsfortheSystemStateDistributionsofMulti-Statek-out-of-nSystems117HisahiYamamotoandTomoakiAkiba1Introduction..........................1172DefinitionfortheMulti-statek-out-of-nSystems.....1193TheoremandAlgorithmfortheSystemStateDistributionsoftheGeneralizedMulti-Statek-out-of-n:FSystem....1214TheoremandAlgorithmfortheSystemStateDistributionsoftheGeneralizedMulti-Statek-out-of-n:GSystem....1285Evaluations..........................1335.1OrdersofComputingTimeandMemorySize...1335.2TheResultofNumericalExperiments.......1356Conclusions..........................1378.Multi-stateComponentsAssignmentProblemwithOptimalNetworkReliabilitySubjecttoAssignmentBudget139Yi-KueiLinandCheng-TaYeh1Introduction..........................1392Assumptions.........................1413SFNunderaComponentsAssignment...........1413.1NetworkReliabilityEvaluation...........1423.2Generatealld-MPs.................1434ProblemFormulation.....................144
10September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookxReliabilityModelingwithApplications5ProposedAlgorithm.....................1446NumericalExperiments...................1476.1Example1......................1476.2Example2......................1497Conclusions..........................1539.ReliabilityAnalysisofaServerSystemwithReplicationSchemes157MitsutakaKimura1Introduction..........................1572ServerSystemwithReplicationusingJournalingFiles..1592.1ReliabilityQuantities................1592.2OptimalPolicy....................1643ServerSystemwithReplicationBufferingRelayMethod.1683.1ReliabilityQuantities................1683.2OptimalPolicy....................1734Conclusions..........................175ComputerSystems17710.Two-DimensionalSoftwareReliabilityGrowthModels179ShinjiInoueandShigeruYamada1Introduction..........................1792StochasticQuantitiesforTwo-DimensionalModeling...1813Two-DimensionalSoftwareReliabilityModeling......1823.1Weibull-TypeTwo-DimensionalSRGM......1823.2BinomialTwo-DimensionalSRGM.........1834ParameterEstimation....................1855ModelComparison......................1876NumericalExamples.....................1907ConcludingRemarks.....................19111.HybridCoordinatedCheckpointingTechniqueUsingIncrementalSnapshots195MamoruOhara,MasayukiArai,SatoshiFukumotoandKazuhikoIwasaki1Introduction..........................195
11September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookContentsxi2ProposedTechnique:HybridCoordinatedCheckpointingUsingIncrementalSnapshots................1973EvaluationModel.......................1993.1StateSavingOverhead...............2003.2RecoveryCost....................2024NumericalExamplesandDiscussions............2045ConcludingRemarks.....................21112.PredictingEffortandErrorsforEmbeddedSoftwareDevelopmentProjectsbyUsinganArtificialNeuralNetwork215KazunoriIwata,SayoriMaejiandToshioNakagawa1Introduction..........................2152SoftwareProjectManagementandIssues.........2163SoftwareDevelopmentProcessesandSelectionofData..2173.1DataSetsforCreatingModels...........2184EffortandErrorPredictionModels.............2194.1AnArtificialNeuralNetworkModel........2194.2NewNormalizationofData.............2214.3MultipleRegressionAnalysisModel........2235EvaluationExperiment....................2245.1EvaluationCriteria.................2245.2DataUsedinEvaluationExperiment.......2245.3ResultsandDiscussion...............2256Conclusion..........................22713.OptimalCheckpointTimesforDatabaseSystems231KenichiroNaruseandSayoriMaeji1Introduction..........................2312RandomCheckpointModels.................2322.1PerformanceAnalysis................2343Database-nodeModels....................2383.1PerformanceAnalysis................2394Conclusion..........................24514.PeriodicandRandomInspectionsforaComputerSystem249MingchihChen,XufengZhaoandSyoujiNakamura1Introduction..........................249
12September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookxiiReliabilityModelingwithApplications2ModelI............................2512.1PeriodicInspection.................2512.2RandomInspection.................2522.3ComparisonofPeriodicandRandomInspections2542.4NthRandomInspection..............2573ModelII............................2583.1PeriodicandRandomInspections.........2583.2ComparisonofPeriodicandRandomInspections2603.3NthRandomInspectionI.............2623.4NthRandomInspectionII.............2644Conclusions..........................266ReliabilityApplications26915.DynamicFaultTreeAnalysis271TetsushiYugeandShigeruYanagi1Introduction..........................2712DynamicGates........................2723AnalysisofDynamicFaultTree...............2733.1MarkovAnalysis...................2743.2AlgebraicApproach.................2763.3BayesianNetwork..................2773.4MonteCarloandPetri-net.............2783.5RepairableDynamicFaultTree..........2784AlgebraicExpressionofDFT................2784.1TemporalFunctionsandOperators........2784.2MinimalCutSequencesSet.............2804.3SequenceProbability................2834.4TopEventProbabilityofDFT...........2865Conclusion..........................28816.ReliabilityAnalysisandModelingTechniqueforanOpenSourceSolution291YoshinobuTamuraandShigeruYamada1Introduction..........................2912ModelingTechnique.....................2922.1StochasticDifferentialEquation..........292
13September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookContentsxiii2.2JumpDiffusionProcess...............2943ParameterEstimation....................2953.1MethodofMaximum-likelihood..........2953.2EstimationofJump-diffusionParameters.....2964SoftwareReliabilityAssessmentMeasures.........2974.1StochasticDifferentialEquationModel......2974.2Jump-diffusionModel................2975NumericalIllustrations....................2985.1DataforNumericalIllustrations..........2985.2ReliabilityAssessmentResults...........2986ConcludingRemarks.....................30417.MaintenanceModelsofMiscellaneousSystems307KodoIto1Introduction..........................3072OptimalMaintenancePolicyforaDamageSystemwithRepair.............................3092.1Model1........................3102.2Model2........................3133ComparisonofThreeCumulativeDamageModels....3153.1ThreeModels....................3153.2OptimalReplacementPolicies...........3174OptimalImperfectMaintenanceofAircraft........3184.1Model1........................3204.2Model2........................3245Conclusions..........................32818.ReliabilityAnalysisofaSystemConnectedwiththeRadioLink331MitsuhiroImaizumi1Introduction..........................3312Model1............................3322.1ModelandAnalysis.................3322.2OptimalPolicy....................3363Model2............................3373.1ModelandAnalysis.................3373.2OptimalPolicy....................3384Model3............................338
14September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookxivReliabilityModelingwithApplications4.1ModelandAnalysis.................3384.2OptimalPolicy....................3405NumericalExample......................3416Conclusion..........................344StudiesonReliabilityandMaintenance34719.StudiesonReliabilityandMaintenance349ToshioNakagawa1Introduction..........................3492PublicationList........................3502.1AcademicMonographs...............3502.2BookChapters....................3502.3Papers........................352
15September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookPART1MaintenancePolicies1
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17September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookChapter1ADynamicProgrammingApproachforSequentialPreventiveMaintenancePolicieswithTwoFailureModesHiroyukiOkamura1,TadashiDohi1andShunjiOsaki21DepartmentofInformationEngineering,GraduateSchoolofEngineering,HiroshimaUniversity,1–4–1Kagamiyama,Higashi-Hiroshima739-8527,Japan2FacultyofInformationSciencesandEngineering,NanzanUniversity,Seto489-0863,Japan1IntroductionStochasticpreventivemaintenanceproblemconsistsofformalism(model-ing)andalgorithm(optimization)forrealapplications.SincetheseminalcontributionbyBarlowandHunter[BarlowandHunter(1965)],ahugenumberofstochasticpreventivemaintenanceproblemshavebeendiscussedfrombothviewpointsofmodelingandoptimization.Sequentialpreventivemaintenancepoliciestodetermineaperiodicmaintenanceschedulesmayberegardedasthemostcriticalbutcomplexonesbecausetheyarere-ducedtononlinearoptimizationproblemswithmultipledecisionvariables.FirstNguyenandMurthy[NguyenandMurthy(1981)]considertwoape-riodicpreventivemaintenancemodelswith/withoutminimalrepairs,andextendtheBarlowandHuntermodel[BarlowandHunter(1965)].Nak-agawa[Nakagawa(1986)]independentlyconsidersthesimilarmodelandextendsitinthesubsequentpaper[Nakagawa(1988)].SincetheabovepapersbyNakagawa[Nakagawa(1986,1988)],thestochasticpreventivemaintenancemodelswithaperiodicmaintenanceschedulesarecalledthe3
18September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book4ReliabilityModelingwithApplicationssequentialpreventivemaintenancemodels,andhaveattractedmuchatten-tionsfrommanyauthors.Linetal.[Linetal.(2000)]extendNakagawamodels[Nakagawa(1986,1988)]intermsofagereductioneffectandhazardrateadjustment.Thesameauthors[Linetal.(2001)]alsotakeaccountoftwofailuremodesandextendtheirmodels[Linetal.(2000)].El-FerikandBen-Daya[El-FerikandBen-Daya(2005)]alsoextendNguyenandMurthymodel[NguyenandMurthy(1981)]withadifferentpolicyfromNakagawa[Nakagawa(1988)]andLinetal.[Linetal.(2001)].Kimetal.[Kimetal.(2007)]consideraproblemtodetermineboththeoptimalnumberofpreventivemaintenanceanditsassociatedtimesequence.SheuandLiou[SheuandLiou(1995)]andSheuandChang[SheuandChang(2002)]formulatethedifferentsequentialpreventivemaintenanceproblemswiththesamelineasNguyenandMurthy[NguyenandMurthy(1981)]andNakagawa[Nakagawa(1986,1988)].Recently,NakagawaandMizutani[NakagawaandMizutani(2009)]giveamodificationofNakagawamodel[Nakagawa(1986,1988)]withafiniteplanningtimehorizon.Itisworthmentioningthatthesepreventivemaintenancepoliciesarenotimpracticalmodels.AgoodillustrativeexampleofthesequentialpreventivemaintenancepolicyisgivenbyJayabalanandChaudhuri[JayabalanandChaudhuri(1992)],wheretheyapplyittothemaintenanceproblemofbusenginesinalargetransportnetworkwithmorethan2500buses.Inthisway,considerableattentionshavebeenpaidtothesequentialpreventivemaintenancemodels.However,themainconcerndevotedintherelatedworkwasjustthemodeling,butnotthecomputationalgorithm.Morespecifically,theunderlyingoptimizationproblemsforthesequentialpreventivemaintenancepoliciescanbereducedtononlinearoptimizationproblemswithmultipledecisionvariables.Inalmostallpapers,itisshownthattheoptimaltimesequenceofpreventivemaintenancemustsatisfythefirst-orderconditionofoptimality,butnoeffectivecomputationalgorithmsarenotdeveloped.Itshouldbesurprisedtoseethattheaboverelatedpaperstothesequentialpreventivemaintenancepoliciesgiveverysmalltoyexerciseandneversolvearealisticlevelofproblemwithmorethan100timepoints.Inthispaper,wetakeanexamplemodelwithtwodifferentfailuremodescorrespondingtotheminimalrepairandreplacement,andformulatethisgeneralizedproblembymeansofthedynamicprogramming(DP).Theresultingalgorithmgivesaneffectivealgorithmtocomputetheaperiodicoptimalmaintenanceschedulewhichminimizestherelevantexpectedcostrate,andisindependentofthekindofmodel.So,theproposedcomputation
19September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookADynamicProgrammingApproachforSequentialPreventiveMaintenancePolicies5schemeprovidesaunifiedframeworktodeterminethesequentialpreventivemaintenancepolicies.2SequentialImperfectPMNguyenandMurthy[NguyenandMurthy(1981)]andNakagawa[Naka-gawa(1986,1988)]proposethesequentialpreventivemaintenance(PM)policies.UnliketheperiodicPMpolicieswithequidistantPMtimeperiod,thesequentialPMpoliciesallowtheaperiodicPMtimesequenceatwhichthesystemshouldundergopreventivemaintenanceoptimally.Considerasystemunderminimalrepairsandreplacement.Thesystemhastwofailuremodes:TypeIandTypeIIfailures,whereTypeIfailureisanerrorofanysystemcomponentandcanbefixedbyaminimalrepair,andTypeIIfailureisafatalerrorandisrepairedbyonlycorrectivereplacement.Inthispaper,weassumethattwotypesoffailureoccurindependently.Moreprecisely,thesystemundergoespreventivemaintenanceateachtimepoints,t1 20September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book6ReliabilityModelingwithApplicationsLetSk(tk|tk−1)andTk(tk|tk−1)betheexpectedtotalcostincurredinthek-thperiodandtheexpectedtimelengthofthek-thperiod,providedthatthek−1-standk-thpreventivemaintenancesarepreformedattk−1andtk,respectively.Thenwehave∫tk−tk−1S(t|t)=chI(t|t)RII(t|t)dtkkk−11kk−1kk−10+cRII(t−t|t)2kkk−1k−1()+c1−RII(t−t|t),(3)4kkk−1k−1k=1,...N−1,∫tN−tN−1S(t|t)=chI(t|t)RII(t|t)dtNNN−11NN−1NN−10+cRII(t−t|t)3NNN−1N−1()+c1−RII(t−t|t),(4)4NNN−1N−1∫tk−tk−1T(t|t)=RII(t|t)dt,(5)kkk−1kk−10k=1,...N,whereRII(t|t)isthereliabilityfunctionforTypeIIfailureinthek-thkk−1period:(∫t)RII(t|t)=exp−hII(s|t)ds,0≤t 21September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookADynamicProgrammingApproachforSequentialPreventiveMaintenancePolicies7themodelrepresentsthehazardratePMandagereductionPMmodels[Linetal.(2000)].Inaddition,whenthefailureratehI(t|t)isreplacedkk−1withhI+III(t|t)=hI(t|t)+hIII(t+t),(9)kk−1kk−1k−1themodelcorrespondstothesequentialPMwithunmaintainablefailuremode(TypeIIIfailure)[Linetal.(2001)].3DPAlgorithmSincetheexpectedcostrateforeachpolicyisgivenasafunctionofNandN,theoptimizationproblemisreducedtoanon-linearprogrammingtoobtainminN,NC(N,N),giventhenumberoftotalmaintenancesN.Itisworthnotingthatthereisnoeffectivealgorithmtofindtheoptimalpair(N∗,∗)simultaneously.Hencethetotalnumberofpreventivemain-Ntenancesmustbecarefullyadjustedaccordingtoanyheuristicmanner.Instead,wefocusonfindingtheoptimalmaintenanceschedule∗underNafixedN.InthecaseofafixedN,themostpopularmethodtofindtheoptimalmaintenanceschedulemightbeNewton’smethodoritsiterativevariants.However,sinceNewton’smethodisageneral-purposenon-linearoptimizationalgorithm,itmaynotoftenworkwelltosolvetheminimiza-tionproblemwithmanyparameters.Inourminimizationproblem,thedecisionvariablesN={t1,...,tN}havetheconstraintt1<··· 22September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book8ReliabilityModelingwithApplicationsbyWk(tk|tk−1,J1,Jk+1,ρ)=Sk(tk|tk−1)−ρTk(tk|tk−1)()+JRII(t−t|t)+J1−RII(t−t|t),(12)k+1kkk−1k−11kkk−1k−1WN(tN|tN−1,J1,ρ)=SN(tN|tN−1)−ρTN(tN|tN−1)+J1.(13)Intheaboveequations,Jk,k=1,...,N,arecalledtherelativevaluefunctions.Equations(10)and(11)arenecessaryandsufficientconditionsoftheoptimalmaintenanceschedule.Thatis,theproblemcanbereducedintofindingthemaintenanceschedulewhichsatisfiestheoptimalityequations.InthelonghistoryoftheDPresearch,thereareacoupleofalgorithmstosolvetheoptimalityequations.Inthispaper,weapplythepolicyitera-tionschemetoderivetheoptimalmaintenanceschedule.Ouralgorithmistwofold:thepolicyimprovementundergivenrelativevaluefunctionsandthecomputationofrelativevaluefunctionsunderamaintenanceschedule.Thesetwostepsarerepeatedlyexecuteduntilthemaintenancescheduleconverges.Inthepolicyimprovement,wefindanewmaintenancesched-ulebasedonthefollowingfunctionsundergivenrelativevaluefunctionsJ1,...,JN:Wk(tk|tk−1,J1,Jk+1,ρ),k=1,...,N−1(14)andWN(tN|tN−1,J1,ρ).(15)However,whenJ1,...,JNareconstants,theabovefunctionsarenotalwaysconvexwithrespecttodecisionvariablestk.Thusourpolicyimprovementalgorithmisbasedonthefollowingcompositefunctionsfortwosuccessiveperiods,insteadofWk(·|·):W˜k(tk|tk−1,tk+1,J1,Jk+2,ρ)=Wk(tk|tk−1,J1,Wk+1(tk+1|tk,J1,Jk+2,ρ),ρ),(16)tk−1≤tk≤tk+1,k=1,...,N−2,W˜N−1(tN−1|tN−2,tN,J1,ρ)=WN−1(tN−1|tN−2,J1,WN(tN|tN−1,J1,ρ),ρ),(17)tN−2≤tN−1≤tN.Theabovecompositefunctionsareconvexintherespectiverangestk−1≤tk≤tk+1,k=1,...,N−1.Inaddition,thefunctionWN(tN|tN−1,J1,ρ)isalsoconvexintherangetN−1≤tN<∞.Thusitispossibletofind 23September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookADynamicProgrammingApproachforSequentialPreventiveMaintenancePolicies9theimprovedmaintenanceschedulebyperformingtheone-dimensionalop-timizationforeachperiodofpreventivemaintenance.Underagivenmaintenanceschedulet1,...,tN,thecomputationstepgivescorrespondingrelativevaluefunctionsandρbysolvingthefollowinglinearsystem:Mx=b,(18)where−RII(t−t|t)ifi=jandj̸=N,iii−1i−11ifi=j+1,[M]i,j=(19)Ti(ti|ti−1)ifj=N,0otherwise,x=(J,...,J,ρ)′,(20)2Nb=(S(t|t),...,S(t|t))′.(21)110NNN−1InEq.(19),[·]i,jdenotesthe(i,j)-elementofmatrix,andtheprime(′)representstransposeofvector.Theabovelinearsystemcomesfromtheoptimalityequations(10)and(11)directly.NotethatJ1=0,sincewearehereinterestedintherelativevaluefunctionJiandρ.Finally,wederivetheDPalgorithmtoderivetheoptimalmaintenancescheduleasfollows.•Step1:Giveinitialvaluesk:=0,t0:=0,(0)(0)(0)N:={t1,...,tN}.•Step2:ComputeJ(k),...,J(k),ρ(k)forthelinearsystem(18)1N(k)underthemaintenanceschedule.N•Step3:Solvethefollowingoptimizationproblems:t(k+1):=argmaxW˜(t|t(k),t(k),J(k),J(k),ρ(k)),iii−1i+11i+2(k)(k)t≤t≤ti−1i+1i=1,...,N−2,t(k+1):=argmaxW˜(t|t(k),t(k),J(k),ρ(k))N−1N−1N−2N1(k)(k)t≤t≤tN−2Nt(k+1):=argmaxW(t|t(k),J(k),ρ(k)).NNN−11(k)t≤t<∞N−1 24September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book10ReliabilityModelingwithApplications(k+1)(k)•Step4:Foralli=1,...,N,if|ti−ti|<δ,stopthealgo-rithm,whereδisanerrortolerance.Otherwise,letk:=k+1andgotoStep2.InStep3,anarbitraryoptimizationtechniquecanbeapplied.Sincethecompositefunctionsareconvexfunctionshavingauniquesolutionintheranges[ti−1,ti+1),i=1,...,N−1,itisnotsodifficulttocalculatetheoptimalpreventivemaintenancetime.Infact,thegoldensectionmethodiseffectivetofindthesolution.4NumericalExamplesWefirstconsiderthecasewherethefailurerateforTypeIfailureisgivenbythefollowingWeibull-typefailurerate:hI(t|t)=αβtβk−1,(22)kk−1kkwhere1/α=100×0.81k−1andβ=2.0.Theotherparametersarekkc1=1.0,c2=3.0,c3=100.0andc4=1000.0.Theseparametersarecitedfrom[Nakagawa(1986)].WeinvestigatethesensitivityofthefailurerateforTypeIIfailureontheoptimalPMsequenceandtheminimumcost.Table1presentstheoptimalPMtimingwhenthetotalnumberofPMsisfixedunderhII(t|t)=0,i.e.,RII(t|t)=1.Asmentionedbefore,kk−1kk−1ourmodelisreducedtoNakagawa’smodelwhenRII(t|t)=1.Fromkk−1thetable,theoptimalnumberofPMsisN=11,whichminimizesthetotalcostrate.In[Nakagawa(1986)],theoptimalPMsequenceinthecaseofN=11waspresented,anditisexactlysameasourresult.Thatis,ourDPalgorithmcansolvethesequentialPMpolicystably.Figure1illustratestheoptimalPMsequencesforN=2through16.Inthefigure,x-axisrepresentsPMtimingandtheoptimalPMsequencewasplottedaspointsonthehorizontalline.Fromthefigure,itcanbefoundedthatthetimeintervalofPMsbecomesmonotonicallydecreasingsequenceforallthecases.Moreover,whenwefocusonthefirstPMtiming,itdecreasesasthenumberoftotalPMsincreasesfromN=2toN=11,butitdecreasesfromN=11toN=16.Inthiscase,theoptimalnumberoftotalPMsisN=11.ThetendencyofPMsequencesarechangedattheoptimalnumberofPMs.NextweinvestigatethecasewherethetypeIIfailurerateisgivenbyaconstant;hII(t|t)=λ.(23)kk−1 25September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookADynamicProgrammingApproachforSequentialPreventiveMaintenancePolicies11Table1OptimalsequentialPMtimingforN=8,9,10,11,12,13,14withouttypeIIfailure.N891011121314t153.152.652.452.352.452.552.8t296.195.394.894.794.895.095.5t3131.0129.8129.2129.1129.2129.5130.1t4159.2157.8157.0156.9157.0157.3158.2t5182.1180.4179.6179.4179.6179.9180.9t6200.6198.8197.9197.6197.8198.3199.3t7215.6213.7212.7212.4212.6213.1214.2t8227.8225.7224.6224.4224.6225.1226.2t9235.5234.4234.1234.3234.8236.0t10242.2241.9242.2242.7243.9t11248.3248.6249.1250.3t12253.7254.3255.5t13258.5259.8t14263.2C(N,N)1.0621.0531.0481.0471.0481.0511.056N=2N=3N=4N=5N=6N=7N=8N=9N=10N=11N=12N=13N=14N=15N=16050100150200250300PMtimeFig.1OptimalPMsequencesforfixedNwithoutTypeIIfailure.Figures2through7depicttheoptimalPMsequencesforrespectivetypeIIfailureratesλ=1/100,1/200,1/300,1/400,1/500,1/1000withafixedN.AlsoTable2presentstheminimumcostratesforeachoptimalPMsequencewithafixedN.ThelastcolumnindicatestheminimumcostratesinthecasewheretheTypeIIfailuredoesnotoccur,i.e.,hII(t|t)=0.Thekk−1 26September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book12ReliabilityModelingwithApplicationsN=2N=3N=4N=5N=6N=7N=8N=9N=10N=11N=12N=13N=14N=15N=16050100150200250300PMtimeFig.2OptimalPMsequencesforfixedN(λ=1/100).N=2N=3N=4N=5N=6N=7N=8N=9N=10N=11N=12N=13N=14N=15N=16050100150200250300PMtimeFig.3OptimalPMsequencesforfixedN(λ=1/200).asteriskmeanstheoptimalnumberofPMsminimizingthecostrates.Foreverycase,weapplytheDPalgorithmtoobtaintheoptimalPMsequences.EvenifthenumberofPMsislarge,e.g.,N=16,wecangettheoptimalPMsequencesstably. 27September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookADynamicProgrammingApproachforSequentialPreventiveMaintenancePolicies13N=2N=3N=4N=5N=6N=7N=8N=9N=10N=11N=12N=13N=14N=15N=16050100150200250300PMtimeFig.4OptimalPMsequencesforfixedN(λ=1/300).N=2N=3N=4N=5N=6N=7N=8N=9N=10N=11N=12N=13N=14N=15N=16050100150200250300PMtimeFig.5OptimalPMsequencesforfixedN(λ=1/400).Fromthefigures,evenwhentheTypeIIfailureoccurs,theoptimalPMsequencehasthesimilartendencyasthecasewheretheTypeIIfailuredoesnotoccur.Also,thePMtimingtendstobeearlierinthecasewhereTypeIIfailurerateishigh;λ=1/100.Moreover,itcanbeseenthatthe 28September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book14ReliabilityModelingwithApplicationsN=2N=3N=4N=5N=6N=7N=8N=9N=10N=11N=12N=13N=14N=15N=16050100150200250300PMtimeFig.6OptimalPMsequencesforfixedN(λ=1/500).N=2N=3N=4N=5N=6N=7N=8N=9N=10N=11N=12N=13N=14N=15N=16050100150200250300PMtimeFig.7OptimalPMsequencesforfixedN(λ=1/1000).minimumcostratestronglydependonTypeIIfailureratefromthetable.Inaddition,inthecaseofthehighTypeIIfailurerate,theoptimalnumberofPMsisbiggerthantheothercases. 29September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookADynamicProgrammingApproachforSequentialPreventiveMaintenancePolicies15Table2Minimumcostrates.N1/1001/2001/3001/4001/5001/1000w/oTypeII211.0096.2364.6563.8673.3942.4511.509310.8516.0554.4683.6763.2022.2561.311410.7665.9574.3673.5743.1002.1521.206510.7165.8994.3063.5133.0382.0891.143610.6835.8614.2683.4742.9982.0501.104710.6625.8364.2423.4482.9732.0241.078810.6475.8194.2253.4312.9562.0081.062910.6375.8084.2143.4212.9461.9981.0531010.6305.8014.2083.4142.9391.9921.0481110.6255.7974.2043.4112.9371.9901.047*1210.6225.7954.203*3.410*2.936*1.990*1.0481310.6205.794*4.2033.4112.9371.9921.0511410.6205.7954.2053.4132.9401.9961.0561510.619*5.7964.2073.4172.9442.0011.0621610.6195.7984.2103.4212.9482.0061.0685ConcludingRemarksInthispaper,wehaveconsideredasequentialpreventivemaintenancemodelwithminimalrepairandreplacement.Themodelunderconsidera-tionisanextensionofNakagawa[Nakagawa(1986,1988)]byintroducingtwodifferentfailuremodes.Toderivetheoptimalpreventivemaintenanceschedulewhichminimizestheexpectedcostrate,wehavedevelopedaDP-basedalgorithmwhichistwofold;policyimprovementandcomputationofrelativevaluefunctions.Inparticular,wehaveappliedcompositefunctionsfortwosuccessiveperiodsofpreventivemaintenancetoderivetheimprovedmaintenanceschedule.AcknowledgmentTheauthorsaregratefultoProf.ToshioNakagawawhostimulatedtheirresearchinterestsinpreventivemaintenancemodelingthroughhismanypapers.ThisresearchwaspartiallysupportedbytheMinistryofEduca-tion,Science,SportsandCulture,Grant-in-AidforScientificResearch(C),GrantNo.23500047(2011–2013)andGrantNo.23510171(2011–2013).ReferencesBarlow,R.E.andHunter,L.C.(1965).Optimumpreventivemaintenancepoli-cies,OperationsResearch8,pp.90–100. 30September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book16ReliabilityModelingwithApplicationsEl-Ferik,S.andBen-Daya,M.(2005).Age-basedhybridmodelforimperfectpreventivemaintenance,IIETransactions38,pp.365–375.Jayabalan,V.andChaudhuri,D.(1992).Sequentialimperfectpreventivemainte-nancepolicies:acasestudy,MicroelectronicsandReliability32,pp.1223–1229.Kim,H.S.,Kwon,Y.S.andPark,D.H.(2007).Adaptivesequentialpreven-tivemaintenancepolicyandBayesianconsideration,CommunicationsinStatistics–TheoryandMethods36,pp.1251–1269.Lin,D.,Zuo,M.J.andYam,R.C.M.(2000).Generalsequentialimperfectpre-ventivemaintenancemodels,InternationalJournalofReliability,QualityandSafetyEngineering7,pp.253–266.Lin,D.,Zuo,M.J.andYam,R.C.M.(2001).Sequentialimperfectpreventivemaintenancemodelswithtwocategoriesoffailuremodes,NavalResearchLogistics48,pp.172–183.Nakagawa,T.(1986).Periodicandsequentialpreventivemaintenancepolicies,JournalofAppliedProbability23,536–542.Nakagawa,T.(1988).Sequentialimperfectpreventivemaintenancepolicies,IEEETransactionsonReliability37,295–298.Nakagawa,T.andMizutani,S.(2009).Asummaryofmaintenancepoliciesforafiniteinterval,ReliabilityEngineeringandSystemSafety94,pp.89–96.Nguyen,D.G.andMurthy,D.N.P.(1981).Optimalpreventivemaintenancepoliciesforrepairablesystems,OperationsResearch29,pp.1181–1194.Sheu,S.-H.andChang,T.-H.(2002).Generalizedsequentialpreventivemainte-nancepolicyofasystemsubjecttoshocks,InternationalJournalofSystemsScience33,pp.267–276.Sheu,S.-H.andLiou,C.-T.(1995).Ageneralizedsequentialpreventivemain-tenancepolicyforrepairablesystemswithgeneralrandomminimalrepaircosts,InternationalJournalofSystemsScience26,pp.681–690. 31September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookChapter2SelectiveMaintenanceforComplexSystemsConsideringImperfectMaintenanceEfficiencyMayankPandey1,YuLiu1,2,MingJ.Zuo1,21DepartmentofMechanicalEngineering,UniversityofAlberta,Edmonton,Alberta,T6G2G8,Canada2SchoolofMechanical,Electronic,andIndustrialEngineering,UniversityofElectronicScienceandTechnologyofChina,Chengdu,Sichuan611731,China1IntroductionAllequipmentandsystemsdeterioratewithageandusage.Maintenanceisrequiredtobeperformedonarepairablesystemtoimprovetheoverallsystemreliabilityandavailability.Iftimelymaintenanceisnotperformed,thesystemmayfail,leadingtohugecostsassociatedwiththefailureandcorrectiveactionsafterward.Correctivemaintenanceisperformedafterthefailurerealizationandaimstomakethesystemperformthedesiredfunctionsaftermaintenance.Correctivemaintenanceisexpensive,henceitisimportantthatafailureisprevented.Preventivemaintenance(PM)isperformedatpre-specifiedintervalsoraspersomecriteriasuchthatthesystemreliabilityisincreasedandfailureisavoided.UndesiredfailureandcorrectivemaintenancecostsaresavedusingPM.Themaintenancecanbecategorizedasfollows:1)Perfectmaintenance/repairorreplacement:Replacementofasystemwhetheritisworkingorfailed,restoresittoasgoodasnew(AGAN)condition.Uponperfectmaintenance,thefailureintensityfunctionofacomponentisthesameasanewcomponent.17 32September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book18ReliabilityModelingwithApplications2)Minimalrepair:Iffailureintensityofacomponentafterrepairisthesameasithadwhenitfailed,therepairactioniscalledminimalrepair.Thesystemoperatingstateisasbadasold(ABAO)afterminimalre-pair.Changingtheheadlightofatruckcouldbeanexampleofminimalrepairbecauseitdoesnotchangetheoverallfailureintensityofthetruck.3)Imperfectmaintenance/repair:Traditionally,itisassumedthatmain-tenancebringsasystembacktoasgoodasnew(AGAN)orasbadasold(ABAO)condition.However,maintenancecanrestoreasystemtoastatesomewherebetweenAGANandABAOconditions.Suchmainte-nance/repairactionsarecalledimperfectmaintenance/repair.Replac-ingonlyafewpartsofasystemcanbeoneexample.Inpractice,PMlengthenstheusefullifetimeofasystembyreducingtheoccurrenceoffailure.Oneofthekeycharacteristicsofamaintenancemodelistheeffectofdifferentkindsofmaintenanceontheeffectiveageand/orhazardrateofthesystem.ModelingtheeffectivenessofPMiswidelystudiedandreviewedintheliteratures[BarlowandHunter(1960);PhamandWang(1996);Dekkeretal.(1997);DekkerandScarf(1998);Wang(2002);DoyenandGaudoin(2004);DesaiandMital(2006)].Withtheintroductionofimperfectmaintenancealongwiththetraditionalre-placementandminimalrepairmodels,differentapproachesareproposedforimperfectmaintenancemodeling.OnecommonapproachistoassumethatPMisequivalenttominimalrepairwithprobabilitypandreplacementwithprobability1−p[ChanandDowns(1978);Nakagawa(1979);MurthyandNguyen(1981);BrownandProschan(1983);SheuandLiou(1995)].Anotherpopularapproachistospecifytheeffectofmaintenanceontheeffectiveageand/orhazardrateofthesystem[LieandChun(1986);Nak-agawa(1986,1988);Linetal.(2001);Pandeyetal.(2013a)].Arecentcomprehensivereviewofimperfectmaintenancemodelsisdocumentedin[PhamandWang(1996);WuandZuo(2010)].Itisnoteworthythatforanyspecificengineeredsystem,modelvalidationandselectionshouldbeconductedtochoosethebestimperfectmaintenancemodelamongallpos-siblecandidates[Liuetal.(2012)].Intheensuingparagraphs,webrieflyreviewtheimperfectmaintenancemodelsthatwillbeusedinthischapter.[LieandChun(1986)]and[Nakagawa(1986)]introducedadjustment/improvementsinthehazardrateandeffectiveageafterPM.[Naka-gawa(1988)]usedadjustment/improvementfactorsforthehazardrateandeffectiveagetosolvesequentialPMproblemsandproposedthat: 33September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookSelectiveMaintenanceforComplexSystems19(i)thehazardrateinthenextPMintervalbecomes‘ah(x)whereh(x)isthehazardrateinthepreviousPMinterval.Theadjustmentfactoris‘a≥1’andx≥0representstimeelapsedfromthepreviousPMtime.(ii)iftheeffectiveageofacomponentistrightbeforethePM,thenitreducesto‘bt’rightafterthePM,where0≤b≤1istheimprovementfac-torineffectiveage.Thefirstmodeliscalledthehazardadjustmentmodel,whilethesecondmodeliscalledtheagereductionmodel.ThehazardrateadjustmentmodelassumesthatthehazardraterightafterPMreducestozerobutincreasesmorequicklyinthenextPMintervalascomparedtothepreviousPMinterval.TheagereductionmodelassumesthatPMre-ducestheeffectiveageandrightafterPMitmaybegreaterthanzero.Thehazardrateisafunctionofeffectiveagebeforeandaftermaintenance.[Linetal.(2001)]proposedthatinmoregeneralcases,PMcannotonlyreducetheeffectiveagebutmayalsoincreasethehazardrate.Theycombinedtheagereductionandhazardrateadjustmentmodelandcalleditahybridimperfectmaintenancemodel.Ifthehazardratefunctionfortimet∈(0,t1)isg(t)andPMatmaintenancebreak,(t1,t2)changesthehazardratetoh(t)fort∈(t2,t3)(seeFig.1).IftheeffectiveageofthesystembeforeandaftermaintenanceareBandA,respectively,thenthecombinedhybridmodel,whichincludestheeffectofthehazardadjustmentandtheagereduction,canbewrittenas:h(t2+x)=ag(bB+x)(1)where,a≥1and0≤b≤1,andx∈{0,t3−t2}.Whena=1,theabovemodelisthesameastheagereductionmodelandforb=0,itisthesameasthehazardadjustmentmodel.Hence,thehybridimperfectmaintenancemodelcanbeusedtocharacterizetheeffectofPMinageneralmanner.PMconsumestime,humanresources,andhasassociatedcosts.How-ever,inadequatemaintenanceschedulesornonessentialservicesmaywastelimitedmaintenanceresources.ThedecisiontoperformPMbecomesmorecomplicatedwhenasystemiscomposedofseveralcomponents.Optimalallocationofmaintenanceresourcesandselectionofasubsetofmainte-nanceactivitiesthatfulfillthesystemrequirementsaftermaintenanceareveryimportant.ThenumberofPMoptionsavailableforasystemdependsonthePMoptionsavailableforeachcomponentwithinthesystem.Themaintenancedecisionofanyofthecomponentswithinasystemwillaffectthesystemperformance.Itisamajorchallengetoconsidertheeffectofthestochasticprocessesforcomponentsalongwithselectingmaintenanceoptionsforeachcomponentinthemulti-componentsystem.However,it 34September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book20ReliabilityModelingwithApplicationsFig.1HybridimperfectPMmodelmaynotbepossibletoperformalldesirablemaintenanceactivitiesdur-ingamaintenancebreakduetolimitedmaintenanceresourcesliketime,budget,andrepairmanavailability.Inthiscase,asubsetofmaintenanceactionsischosentomakesurethatthedesiredoperationperiodafterthemaintenancebreakissuccessfullycompleted.Thismaintenancepolicyiscalled‘selectivemaintenance’.Formanytypesofequipmentorsystems,breaksbetweensuccessiveoperatingperiods(missions)offerthebestopportunityformaintenance.Someexamplesmayincludemanufacturingequipment,militaryvehicles,powergenerationunits,etc.Manufacturingequipmentmayworkduringtheweekandbemaintainedduringweekends,andsimilarly,powergenera-tionunitsmayworkforthewholeweekandmaintenancecanbeperformedearlySundaymorning.Aircraftmaybemaintainedbetweenflightsandmil-itaryequipmentmaybemaintainedbetweenoperations.Inalloftheabovecases,maintenanceisrequiredtobeperformedbetweenmissionssuchthatthesystemperformssatisfactorilyduringthenextmission.Afterinspec-tion,therecouldbesomepossiblemaintenanceactionstobeperformedduringmaintenancebreaks,e.g.,minimalrepair,differentpreventivemain-tenanceactions,orsystemreplacement.Aftermaintenance,thesystemshouldworkfulfillingthedesiredobjectiveduringthenextmissionuntilthenextscheduledmaintenancebreak.Sinceeachoftheavailablemainte-nanceoptionsconsumessomemaintenanceresources,e.g.timeandcost,theoptimalallocationofresourcesisrequired. 35September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookSelectiveMaintenanceforComplexSystems21Ourprimaryfocusisonexplainingthedifferentselectivemaintenancemodelsconsideringimperfectmaintenance.Thischapterfocusesonmod-elingratherthanstatisticalinference.Wehavetriedourbesttoreviewtheselectivemaintenanceforasystemfairlycompletely;however,somepa-pers,whicharenotincluded,wereeitherconsiderednotrelateddirectlyorwereoverlookedunknowingly.Wehavetodiscussallrelevantpaperswithafocusonthemostrecentworksinthedomainofselectivemaintenancethatdiscussesimperfectrepair/maintenance.Wehavediscussedtheresultsrelatedtotheeffectofimperfectmaintenance/repaironly.Otherresultsirrelevanttoimperfectmaintenance/repairarenotincluded.Thischapterisdividedasfollows:Sec.2brieflyexplainspreviouslit-eraturerelatedtoselectivemaintenanceforsystemswitheitherworkingorfailedstates,i.e.binarystates.Moreexplanationisprovidedforthemostrecentworksonselectivemaintenanceforbinarysystemsdiscussingimperfectmaintenance.Forsomesystems,morethantwoperformancestatesarepossible,thesesystemsarecalledmultistatesystems(MSS).InSec.3,selectivemaintenancestrategiesfortheMSSareexplained.Sec.4summarizestheconclusion.2SelectiveMaintenanceforBinarySystemsTraditionallyasystemisassumedtobeintwostates;workingorfailed.Suchsystemsarecalledbinarysystems.Forbinarysystems,aselectivemaintenanceproblemwasintroducedby[Riceetal.(1998)],whereasys-temwithseries-parallelconfiguration,constantcomponentfailurerates,i.e.exponentialdistributionandonlyonetypeofmaintenanceaction(re-placementoffailedcomponent)wasconsidered.Theyonlyconsideredtimeasaresourceconstraint.Theystudiedamulti-componentseries-parallelsystemasanexampleandmaximizedsystemreliabilityduringthenextmissionsuchthatmaintenanceisperformedwithintheavailabletime.Aseries-parallelsystemisshowninFig.2.InFig.2,thereareSsubsystems(j=1,2,...,S)connectedinaseriesandeachofthejthsubsystemhasnjcomponentsconnectedinparallel.[Riceetal.(1998)]assumedthatallcomponentswithinasubsystemwereidentical.Theyalsoproposedaheuristicthatwasgoodforidenticalcomponentswithinasubsystem.[Cassadyetal.(2001a)]extendedthemodelpresentedby[Riceetal.(1998)]andincludedcostasonemoreresourceconstraintinad-ditiontotime.Byselectingreliability,cost,ortimeastheobjectiveandtheothertwoasconstraints,theydevelopedthreedifferentselectivemain- 36September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book22ReliabilityModelingwithApplications111222n1n2nSFig.2Aseries-parallelsystemtenancemodels.Inthefirstmodel,theymaximizedreliabilityunderthecostandtimeconstraint,inthesecondmodelmaintenancecostwasmini-mizedwithmissionreliabilityandmaintenancetimeasconstraints,andinthethirdmodeltheyminimizedmaintenancetimeundermissionreliabilityandmaintenancecostconstraints.Thefirstmodelisusefulwheresystemreliabilityisimportant.Theymentionedthatforsafety-criticalapplica-tions,e.g.maintainingaircraftorspaceshuttles,itwaslikelythatthefirstmodelwouldbeusedduetotheseverityofasystemfailure.Inaprofitorientedapplication,likemanufacturingsystems,thesecondmodelismorelikelytobeselected.Inserviceorientedapplications,e.g.maintainingcom-puterservers,maintenancetimemaycorrespondtothelossofservice.Inthiscase,thethirdmodelcouldbeused.Further,[Cassadyetal.(2001b)]assumedthatthecomponents’life-timesfollowtheWeibulldistribution.Hence,ratherthanconstantfailureratesusedinpreviousstudies,anincreasingfailureratewasconsideredin[Cassadyetal.(2001b)].Theyalsoincreasedthepossiblemaintenanceactionsforacomponent.Earlier,onlythereplacementoffailedcompo-nentswasapossiblemaintenanceoption;however,[Cassadyetal.(2001b)]includedminimalrepairandreplacementofworkingcomponentsasaddi-tionalmaintenanceoptions.Theylimitedtheirstudytoonlytimeasaresourceconstraint.Later,[SchneiderandCassady(2004)]consideredmultipleseries-parallelsystemssimultaneouslyandcalleditafleet.They 37September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookSelectiveMaintenanceforComplexSystems23usedthemodelof[Riceetal.(1998)]andsolvedselectivemaintenanceproblemsforafleet(consistingofmultiplesystemstogether)performance.Thereareotherworksthataddresstheproblemofimprovingthecompu-tationalefficiencyoftheselectivemaintenanceoptimization.[RajagopalanandCassady(2006)]proposedfourtypesofenumerationmethodstosolveselectivemaintenanceproblemsforaseries-parallelsystem.TheirgoalwastoreduceCPUtimetosolveselectivemaintenanceproblems.However,theyassumedthatallcomponentsinasubsystemaresimilarandonlythereplacementoffailedcomponentswaspossible.Fornon-identicalcompo-nents,non-identicalmaintenancetimefromonecomponenttoanotherinasubsystem,orincreasesinthenumberofmaintenanceoptions,theirheuris-ticbecomesinefficient.Thus,itwasfoundthattheenumerationmethodhadlimitedapplication.[Lustetal.(2009)]alsomentionedthatforasys-temwithalargenumberofcomponents,theselectivemaintenanceprob-lembecomescombinatorialinnatureandtheenumerationmethodwasnolongeruseful.TheyproposedaheuristictogenerateaninitialsolutionanduseditasinputtothebranchandboundprocedureandtheTabusearchalgorithm(anevolutionaryalgorithm).TheyfoundthattheTabusearchprovidedoptimalorveryclosetooptimalsolutionspromptlyascomparedtothebranchandboundmethod.Itwasthefirsttimein[Lustetal.(2009)]thatanevolutionaryapproachwasusedtosolvetheselectivemaintenanceproblem.Thus,anevolutionaryalgorithmwasfoundtobeusefulinsolvingaselectivemaintenanceoptimizationproblem.Later[Zhuetal.(2011)]alsousedtheTabusearchalgorithmwithheuristictosolvetheselectivemaintenanceproblemforaseries-parallelmanufacturingsystem.[Iyoobetal.(2006)]focusedonresourceallocationforsubsequentmis-sionsunderselectivemaintenance.Theyalsocombinedredundancyallo-cationproblemswiththeselectivemaintenancedecisionmaking.[Maillartetal.(2009)]consideredselectivemaintenanceforamulti-missionproblemwithmaintenancepossibleonlyatthebeginningofthefirstmission,whichmakesitsapplicationlimited.[Pandeyetal.(2012)]usedanagebasedimperfectrepairmodelsimilarto[Nakagawa(1988)],butadditionally,theydefinedanagereductionfactorbasedonthecostandeffectiveage.Theymentionedthatwhenacomponentisrelativelynewer,agereductioncanbeachievedusingsomemaintenancecost;however,whenthecomponentbecomesold,morecostisrequiredtoachievethesameamountofagere-duction.Theyfurtherextendedtheirworkin[Pandeyetal.(2013a)]andproposedahybridimperfectmaintenancemodelwheretheimprovementfactorsarebasedontheeffectiveageandthemaintenancecost.Inthe 38September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book24ReliabilityModelingwithApplicationsfollowingsectionsweexplainthemodelusedin[Pandeyetal.(2013a)].Theyusedthefollowingassumptions:1)Thesystemconsistsofmultiple,repairablecomponents.2)Thecomponents,aswellasthesystem,isinabinarystate,i.e.,itiseitherworkingorfailed.3)Afterreplacement,thecomponentis‘asgoodasnew’andifminimalrepairisperformed,itis‘asbadasold’.Maintenanceisalsopossiblesuchthatthecomponenthealthmayliebetweenasgoodasnewandasbadasold,i.e.,maintenancecanbemodeledbyimperfectrepair.4)Limitedresources(budget,repairman,andtime)areavailableandtheamountofresourcesrequiredformaintenanceactivitiesareknownandfixed.Wheneverthesystemcomesinformaintenanceafteramission,adeci-sionistobemadeforeachcomponentregardingmaintenance.Thecom-ponentcanbeineitherworkingorfailedstateafteramission.Ifacompo-nentiinsubsystemjisinworkingstatebeforemaintenance,itisdefinedasYi,j=1,otherwiseYi,j=0.Similarly,aftermaintenanceaworkingstateofcomponent(i,j)isdefinedasXi,j=1,otherwiseXi,j=0.De-pendingontheavailableresourcesandthecomponent’sage,amaintenancedecisionisdeterminedforasystem.Ingeneral,themaintenancequalityimproveswiththeamountofthebudgetinvestedduringmaintenance.Asgivenin[LieandChun(1986)],themaintenancecostusedandtheageofthecomponentaretwoimportantfactorsfordeterminingtheagereductionfactor(b)foracomponent.Basedonthedefinitiongivenin[LieandChun(1986)],[Pandeyetal.(2013a)]formulatedanagereductionfactoras:()m(Bi;j)(CMRi;j(li;j)−Ci;j)1−CR,forYi,j=0,2≤li,j 39September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookSelectiveMaintenanceforComplexSystems25ageof(i,j)beforemaintenanceandm(Bi,j)iscalledthecharacteristicconstant.[Pandeyetal.(2013a)]usedm(Bi,j)toincorporatetheeffectofeffectiveageofthecomponent.Theydefinedm(Bi,j)as:Bi,jBi,jBi,j×R(Bi,j)m(Bi,j)==(∫∞)=∫∞(3)MRLBi;jR(x)dxR(x)dxBi;jR(Bi;j)Ifacomponent’seffectiveageislessthanitsmeanresiduallife(MRL),itisassumedtoberelativelyyounger.However,whenthemeanresiduallifeofthecomponentislessthantheeffectiveageofthecomponent,itissaidthatthecomponentisrelativelyold.Theyoungeracomponentis,thebetteritrespondstothemaintenancebudgetinvested.Inequation(2)useofm(Bi,j)defineshowtherelativeageofthecomponentwillaffecttheamountofbudgetused.2.1HazardAdjustmentFactorAftermaintenance,thehazardratemayalsochangewiththeeffectiveageofacomponent.[Nakagawa(1988);Linetal.(2001);Liaoetal.(2010)]assumedthatthehazardratechangesbyaconstantfactorforacomponentduringeachmaintenance.Inadditiontothetimeofparticularmaintenancebreak,hazardrateaftermaintenancemayalsobeaffectedbythebudgetusedformaintenance.Ifthebudgetissmall,lowimprovementincompo-nenthealthisexpected.Itshazardrateincrementaftermaintenancewillbehighercomparedtothecasewhenalargebudgetisinvolvedinthecompo-nent’smaintenance.Basedontheaboveargument,[Pandeyetal.(2013a)]providedthefollowingformulationforthehazardadjustmentfactor,p()1(Ci;j(li;j)−CMRm(Bi;j)(p−1)+i;jCRi;jforYi,j=0,3≤li,j 40September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book26ReliabilityModelingwithApplications1.21.181.161.141.120.70.91.11.72.12.71.11.08Hazardadjustmentfactor1.060.31.040.151.02100.10.20.30.40.50.60.70.80.91costratio(PMcost/Replacementcost)Fig.3Hazardadjustmentfactorversuscostratiofordifferentvaluesofm(forp=6)componentcanachieveafteramaintenancebreak.Thesmalleristhevalueofp,thelargeristhemaximumallowablehazardadjustmentandvice-versa.Thevalueofpisdeterminedfromthecomponent’shistory.Variationsofthehazardadjustmentfactorfordifferentvaluesofm(Bi,j)areshowninFig.3.Thisfigureshowsthatforasmallercostratio,i.e.,whenasmallerbudgetisusedformaintenanceofacomponent,thehazardratewillin-creasefasteraftermaintenance(highervalueofhazardadjustmentfactor)andvice-versa.Italsoshowsthatforafixedvaluehazardadjustmentfac-tor,theamountofbudgetneededincreasesasthecomponentages(i.e.m(Bi,j)increases).Oncetheagereductionandhazardadjustmentfactorsaredeterminedformaintenancedecisionforeachcomponentinthesystem,itisrequiredtodeterminethesystemreliability.2.2SystemReliabilityEvaluationLetusassumethatpi,j,li;jistheprobabilitythatacomponent(i,j)afterundergoingmaintenanceoptionli,j,finishesthenextmissionsuccessfully.Thisprobabilityshowsthereliabilityofthecomponentforagivenmis-sionduration.IfthelengthofthenextmissionisL,ti,jisthebeginningofthenextmission,thecomponenthazardrateduringthenextmission 41September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookSelectiveMaintenanceforComplexSystems27()hi,j,li;j(ti,j+x)canbeobtainedfromtheequation(1):hi,j,li;j(ti,j+x)=a(Bi,j.li,j)×g(b(Bi,j,li,j)×Bi,j+x),0≤x≤L(5)Theprobabilityofthiscomponentsuccessfullycompletingthenextmissionis:(∫)Lpi,j,li;j=exp−hi,j,li;j(ti,j+x)dx(6)0Thus,thereliabilityofthecomponent(i,j)canbedefinedas:Ri,j,li;j=pi,j,li;j×Xi,j(7)Hence,systemreliabilityforthenextmissioncanbegivenas:∏S∏S∏ni()R(l)=Ri(l)=1−1−Ri,j,li;j(8)i=1i=1j=1wherel=[l1,1,...,li,j,...,ls,ns]isavectorcomprisingthemaintenancede-cisionvariableli,jforallcomponentsinthesystem.2.3SelectiveMaintenanceModelingIfasystemcomesformaintenanceafteramissionwithaknownstateYi,j,effectiveage(Bi,j),andthelifetimedistributionparametersforallcomponents,onlyasubsetofmaintenanceactionscanbeperformedduetolimitedresources.IfthebudgetconstraintduringthemaintenancebreakisgivenbyC0andtheavailablemaintenancedurationislimitedtoT0,theselectivemaintenanceoptimizationproblemtomaximizetheprobabilityofsuccessfullycompletingthenextmissionisdevelopedas:Objective:∏S∏ni()maxR(l)=1−1−Ri,j,li;j(9)i=1j=1Subjectto:C≤C0(10)T≤T0(11) 42September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book28ReliabilityModelingwithApplications{1,ifli,j>1Vi,j=(12)0,otherwise{Vi,j,ifYi,j=0Xi,j=(13)Yi,j,otherwise1≤li,j≤Ni,j(14)Inthisformulation,CisthetotalcostofmaintenanceandTisthetimetoperformmaintenanceforthewholesystem.Furtherdetailstocalcu-latemaintenancecostandtimecanbefoundin[Pandeyetal.(2013a)].Constraints(10),and(11)showthatthelimitedresourcesareavailabletoperformmaintenance,constraints(12),and(13),setthecomponentstateatthebeginningofthenextmission,dependingonthestateattheendofthepreviousmissionandthemaintenanceactionperformed.2.4ResultsBasedontheabovemodel,[Pandeyetal.(2013a)]solvedtheselectivemaintenanceproblemofmulti-componentseries-parallelsystemasshowninFig.4.TheyusedDifferentialEvolution(DE)[Brestetal.(2006)]tosolvetheproblem.(1,1)(2,1)(1,2)(2,2)Fig.4AseriesparallelsystemTheyassumedherethatforintermediatemaintenanceactions,associatedtimeandcostvariedlinearlyas:t=(l−1)×∆tWandc=i,j,li;ji,ji,ji,j,li;j(l−1)×∆cWforY=1andt=TMR+(l−2)×∆tFandi,ji,ji,ji,j,li;ji,ji,ji,jc=CMR+(l−2)×∆cFforY=0.Here∆tW,∆cW,∆tFi,j,li;ji,ji,ji,ji,ji,ji,ji,jand∆tFindicatethetimeandcostrequiredtoincreasetheintermediatei,j 43September17,201314:50SelectiveMaintenanceforComplexSystemsBC:9023-ReliabilityModelingwithApplicationsTable1Systemparameters,maintenancetimeandcost(i,j)αi;jβi;jYi;jBi;jTMRTWR∆tWTFR∆tFCMRCWR∆cWCFR∆cFi;ji;ji;ji;ji;ji;ji;ji;ji;ji;j(1,1)151.5115350.2510.256122121(1,2)151.5120350.2510.255121.75121(2,1)20308240.220.25141.5142(2,2)203115240.220.26151.6151.5292013book 44September17,201314:5030Table2Comparisonwhenonlyreplacement/minimalrepairisusedandwhenimperfectrepair/maintenanceisincluded(To=9units,Co=25units)[Pandeyetal.(2013a)]ReliabilityModelingwithApplicationsComp#(i,j)Withimperfectrepair(Proposedmodel)ReplacementandminimalrepairBC:9023-ReliabilityModelingwithApplicationsonlyli;jTi;jCi;jXi;jAi;jli;jTi;jCi;jXi;jAi;j(1,1)DN00115DN00115(1,2)WR51210WR51210(2,1)IM2.81312.7466MR25120(2,2)DN00115DN00115∑∑∑∑=7.8=25=7=17R(l)0.72930.6140*Ti;j=Timespenton(i,j),Ci;j=Costspenton(i,j),Xi;j=Stateof(i,j)aftermaintenance,1≤li;j≤6forYi;j=1,1≤li;j≤7forYi;j=0,Ai;j=Effectiveageaftermaintenance,DN=Donothing,WR=Replacementofaworkingcomponent,IM=Imperfectmaintenance,MR=Minimalrepair.2013book 45September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookSelectiveMaintenanceforComplexSystems31maintenancelevelbyunityforworkingandfailedcomponents,respectively.TheyusedtheparametersshowninTable1.InTable1,αi,jisthescaleparameterandβi,jistheshapeparame-terfortheWeibulldistribution,Yi,jisthestatebeforemaintenance,Bi,jistheeffectiveagebeforemaintenance,TMRandCMRisthetimeandi,ji,jcosttoperformminimalrepair,TWRandCWRisthetimetoreplacethei,ji,jworkingcomponent(i,j),TFRandCFRisthetimeandcosttoreplacei,ji,jthefailedcomponent(i,j).Theyassumedthatp=8foreachcomponentinthesystem(equation(4)).ForthemissiondurationofL=8timeunits,budgetlimitofC0=25units,andtimeconstraintofT0=9units,[Pandeyetal.(2013a)]comparedtwoscenarios.Inthefirstscenario,onlyreplace-mentandminimalrepairarepossiblemaintenanceoptions,whileinthesecondscenarioimperfectmaintenance/repairisalsopossible,alongwithreplacementandminimalrepair.TheresultsareshowninTable2.FromTable2,itcanbeseenthatinclusionofimperfectmainte-nance/repairasmaintenanceoptionsincreasesthesystemreliabilitybymorethan11%.Hence,itisimportanttoincludetheimperfectmainte-nance/repairasamaintenanceactionforselectivemaintenance.Itprovidesflexibilitytousetheavailableresourcesinanoptimalmannersuchthatsys-temreliabilityismaximized.[Pandeyetal.(2013a)]alsocomparedtheagereductionmodel,hazardadjustmentmodel,andhybridimperfectmodel.Itwasfoundthatfortheaboveproblem,theagereductionmodelgivesthenextmissionsystemreliabilityof0.7324andthehazardadjustmentmodelgivesamissionsystemreliabilityof0.88,respectively.Thesystemreliabil-ityforthehybridmodelis0.7293.Thereasonisthatfortheagereductionmodel,thereisnochangeinthehazardrateandforthehazardratemodel,effectiveagebecomeszeroatthebeginningofthenextmission,hencebothindividualmodelsgivehighersystemreliability.However,whencombined,i.e.hazardrateincrementaswellastheeffectiveagechangeisconsidered,lowersystemreliabilityisachievedforthehybridmodel.Theirresultsweresimilartowhatwasobservedby[Linetal.(2000)]forschedulingPMforasystem.3SelectiveMaintenanceforMulti-stateSystemsForsomesystemsorcomponents,thebinaryassumptiondoesnotreflectthepossiblestatesthateachofthemmayexperience.Theycanperformtheirtaskswithvariousdiscretelevelsofefficiencyknownas‘performancerates’[LisnianskiandLevitin(2003)].Asystemthathasafinitenumber 46September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book32ReliabilityModelingwithApplicationsofperformanceratesvaryingfromperfectoperationtocompletefailureisdefinedasamulti-statesystem(MSS).Forexample,apowergenerationsystemmayhavefourstatesdenotedas0,1,2,and3.Theperformanceratesforthesestatesmaycorrespondtotheoutputsof0MW,30MW,50MW,and80MW,respectively[Massimetal.(2005)].ThereareonlyafewworksforselectivemaintenanceofanMSS.First,[Chenetal.(1999)]workedontheselectivemaintenanceoptimizationforamulti-stateseries-parallelsystem.Theyconsideredallcomponentsandsysteminmultiplestatesandoptimizedcostassociatedwithtransitionfromonestatetoan-otherforeachcomponentandtheoverallsystem.Intheirproblem,tran-sitionprobabilitiesandassociatedcostswereknown.However,theydidnotshowthemaintenanceactionrequiredforacomponentorcomponentsstateaftermaintenance.Theirheuristicwasgoodforasystemwithasmallnumberofcomponentsonly.AnotherworkonselectivemaintenanceforanMSSwasdoneby[LiuandHuang(2010)].Theyassumedthatcomponentswithinasystemwereinabinarystate,i.e.,eitherworkingorfailed;however,thesystemitselfcouldhavemultiplestates.Theimperfectmaintenanceefficiencywasconsideredintheirselectivemaintenance,andacost-maintenanceefficiencyrelationthatconsideredtheagereductionfactoroftheKijimaimperfectmainte-nancemodelasafunctionofassignedmaintenancecostwasestablished.Recently,[Pandeyetal.(2013b)]relaxedbinarycomponentassumptionandsolvedtheselectivemaintenanceoptimizationproblemforanMSSwherecomponentscouldalsoexhibitmultipleperformancelevels.Inthischapter,wewilldiscussbothoftherecentworksof[LiuandHuang(2010)]and[Pandeyetal.(2013b)].3.1SelectiveMaintenanceforMSSwithBinaryComponentsIn[LiuandHuang(2010)],thebasicassumptionsforthestudiedMSSareasfollows:1)TheMSSconsistsofMbinarystatescomponents,andperformanceratesforeachcomponenti(i∈{1,...,M})aredenotedbythesetgi={gi,1,gi,2},wheregi,2(̸=0)isanominalperformancerate,andgi,1=0representsfailure.2)TheMSScanbeconstructedbycomponentsinarbitraryconfiguration,suchasseries-parallel,bridge,complexnetwork,etc. 47September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookSelectiveMaintenanceforComplexSystems333)Atthebeginningofthekthmission,thestatusofcomponentiisrepre-sentedbybinaryvariableXi(k),where{1ifthecomponentiisfunctioningXi(k)=.0ifthecomponentiisinfailurestate4)Ai(k)representstheeffectiveageofcomponentiatthebeginningofthekthmission,andB(k)istheeffectiveageofcomponentiwhentheikthmissionisover.5)ThedurationofthekthmissionisdenotedbyT(k).6)Anymaintenanceactioncanonlybeexecutedduringthebreakbetweentwosuccessivemissions.Inthebreak,thereexistslimitedresources(e.g.cost,time,repairmen,etc.)toperformmaintenance.Decision-makersneedtodeterminehowtoallocatethemaintenanceresourcestoindividualcomponentswiththeaimofrestoringtheentireMSStothestatethatmaximizestheprobabilityofsuccessfullycompletingthesubsequentmission.7)Theprobabilityofthesystemsuccessfullycompletingamissionisde-finedastheprobabilitythattheperformancerateoftheMSSisnotlessthanthedemandlevelduringthewholemission.8)Multiplemaintenanceactionscanbechosenforbothfailed,andfunc-tioningcomponents,includingminimalrepair,corrective/preventivereplacement(orperfectmaintenance),andimperfectmaintenance.Themaintenanceefficiencyrelateswithassignedmaintenancecost(orre-sources),andtheirrelationcanbemeasured.TheefficiencyofimperfectmaintenanceischaracterizedbytheKijimatypeIIagereductionmodel[Kijima(1989)].TheeffectiveageofanycomponentiafterthemaintenancesubsequenttothekthmissionisgivenbyAi(k+1)=bi(k)Bi(k).(15)Ifthebinarystatecomponentifailsduringthekthmission,itseffectiveagewillimmediatelystopincreasingwithchronologicaltimeasshowninCase3inFig.5.Inthisfigure,failureshappen,respectively,atthe2nd,and7thdaysofchronologicaltime;andtheeffectiveageofthecomponentissteadyattheremainingmissiontime.bi(k)(0≤bi(k)≤1)istheagereductionfactorrepresentingmaintenanceefficiency,andasmallerbi(k)meansagreaterimprovementasplottedinFig.5(seeCases1and2wherenofailurehappens). 48September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book34ReliabilityModelingwithApplicationsFig.5ChronologicaltimeversuseffectiveageintheKijimatypeIImodelBasedontheeffectiveagemodel,iftherandomquantityYindicatesthelifetimeofacomponentandt,istheelapsedtimeaftermaintenance,theconditionalsurvivalprobabilityofacomponentafteramaintenanceactivityisgivenby:Pr{t 49September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookSelectiveMaintenanceforComplexSystems35(18).Inthesamefashion,theagereductionfactorasafunctionofthepreventiverepaircostisexpressedas()1pci(k)mibi(k)=1−rp,(19)cippwheremi(mi>0)isacharacteristicconstantthatdeterminestheexactrelationbetweenpreventiverepaircostandthecorrespondingagereductionfactorthrough(19).TheprobabilityofsuccessfullycompletingamissionisdefinedastheprobabilitythattheMSSperformancerateisnotlessthanthemissiondemandlevelduringthewholesinglemissionperiod.TheformulaoftheprobabilityofsuccessfullycompletingasingleconsecutivemissionforanMSSispresentedinSec.3.2.1.Hence,theselectivemaintenancemodelingforanMSSwithbinary-statecomponentsisexpressedas:Objective:maxRS(Wk+1,T(k+1),X(k+1),A(k+1))=∑PJ(T(k+1),X(k+1),A(k+1))(20)GJ(t)≥Wk+1Subjectto:C≤C0(21)Ai(k+1)=bi(k)·Bi(k)(22)Xi∈{0,1}(23)whereA(k+1)={A1(k+1),...,AM(k+1)}isavectorrepresentingtheeffectiveagesofcomponentsatthebeginningofmissionk.X(k+1)={X1(k+1),...,XM(k+1)}isabinary-valuedvectorcontainingthestates,eitherfunctioningorfailed,ofcomponentsatthebeginningofmissionk.A(k+1)andX(k+1)arecompletelydeterminedbythetypeofmainte-nances(i.e.correctiveorpreventive)andtheamountofmaintenancecostperformedonorassignedtoeachcomponent.CisthetotalmaintenancecostfortheMSSafterthelastmission;whereasC0isthemaintenancebudgetduringthemaintenancebreak.ItshouldbenotedthatselectivemaintenanceoptimizationforanMSSwithimperfectmaintenanceisacom-plex,non-linear,continuousprogrammingproblemasshownin(20)-(23).Anexhaustiveexaminationofallpossiblesolutionsisnotrealisticdueto 50September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book36ReliabilityModelingwithApplicationsthecomputationaltimelimitation.Meta-heuristicalgorithms,suchasge-neticalgorithm(GA),differentialevolution(DE),Tabusearch,simulatedannealingalgorithm,andantcolonyoptimization(ACO),canbeusedtosolvetheresultingoptimizationinacomputationalefficientmanner.Acoaltransportationsystemsupplyingcoaltoaboilerinapowersta-tionispresentedhereasanexampletodemonstratetheeffectivenessofourproposedmethod.ThestudiedsystemincludesfivebasicsubsystemsasshowninFig.6.Feeder1(subsystem1)transferscoalfromthebintoconveyor1(subsystem2).Conveyor1transportsthecoalfromfeeder1tothestackerreclaimer(subsystem3)thatliftsthecoaluptotheburnerlevel.Feeder2(subsystem4)thenloadsconveyer2(subsystem5)thattransfersthecoaltotheburnerfeedingsystemoftheboiler.Everysubsystemconsistsofbinarystatecomponents.Theparametersettingsforeachcomponent,e.g.nominalperformancerate(ton/hour),parametersoftheWeibulllifedistribution,maintenancecost,effectiveage,andstatusafterlastmission(thekthmission),aretabulatedinTable3.Theunitsoftime,andcostaredays,and$1,000,respectively.Theuncertaindemandforthe(k+1)thmissionisdistributedasshowninTable4,withtherequireddemandlevels,andtheircorrespondingprobabilities.1116491227510133814Feeder1Conveyor1Stacker-reclaimerFeeder2Conveyor2(Subsystem1)(Subsystem2)(Subsystem3)(Subsystem4)(Subsystem5)Fig.6BlockdiagramofacoaltransportationsystemSupposethedurationofthe(k+1)thmissionisT(k+1)=10days.Al-thoughthesystemisstillfunctioningattheendofthelastmission(thekthmission)withoutanymaintenanceaction,theprobabilityofitsuccessfullythcompletingthenextmission(the(k+1)mission)isonly0.006.GiventhemaintenancebudgetC(k)=$200,000inthebreakafterthekthmission,0onehastooptimallyallocatethemaintenancecosttoeachcomponenttothmaximizetheprobabilityofsuccessfullycompletingthe(k+1)mission. 51September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookSelectiveMaintenanceforComplexSystems37Table3Parametersofcomponents:whereperformancerateisintons/hour,timeisindays,andcostsarein$1,000unitsComp.IDgηβmpcrpmfcrfc0B(k)Y(k)i;2iiiiiiiii1551.5252.5152.52533512802.4382.2202.032424031201.6282.6253.03534504902.6402.2203.235535051451.8281.8254.03422816702.4342.4153.22033617952.5262.8243.03064408802.0282.3202.83552809951.2262.0182.5283381101301.4352.5202.835615011502.8403.2223.032730012751.5352.6252.235422113852.4302.8182.836638114952.2452.2152.6383350Table4MissiondemandsDemand(ton/hour)12090603010Probability0.10.250.350.20.1Thegeneticalgorithmisusedtosearchtheglobaloptimalsolution,andthebestmaintenancestrategyispresentedasScenario1inTable5.Theoptimalallocationsofrepaircostsarelistedinthecolumn“Cost”withtherelatedfixedmaintenancecostinparentheses.FromTable5(Scenario1),onecanseethatallthecomponentsarefunctioningatthestartofthe(k+1)thmission,onlycomponents3and7aresubjectedtocorrectivere-placement,andcomponent5issubjectedtopreventivereplacement.Alloftheotherfailedcomponentsareimperfectlyrepairedbeforethenextmis-sionisexecuted,andsomefunctioningcomponents(6and13)aresubjectedtoimperfectPM.Theprobabilityofsuccessfullycompletingthe(k+1)thmissionis0.77342,andtotalmaintenancecostis$199,880.Theoptimalsolutionforthecasewhereonlyminimalrepair,preventive,andcorrectivereplacement(Scenario2)areconsideredisalsotabulatedinTable5.Theprobabilityofsuccessfullycompletingthemissionis0.7336,andcorrespond-ingtotalmaintenancecostis$199,000.AsshowninTable5,althoughthe 52September17,201314:5038Table5Optimalsolutionsandcomparison[LiuandHuang(2010)]CompIDWithimperfectmaintenance(Sce-Withoutimperfectmaintenance(Sce-nario1)nario2)ActionCostXi(k+1)Ai(k+1)ActionCostXi(k+1)Ai(k+1)ReliabilityModelingwithApplications1DN0135PR15(3)102IC5.33(4)114.2CR32(4)10BC:9023-ReliabilityModelingwithApplications3CR35(3)10MC0(3)1454IC17.5(5)16.82CR35(5)105PR25.0(2)10DN01286IP8.57(3)17.49DN01367CR30(6)10CR30(6)108IC5.84(5)113.23MC0(5)1289DN0138PR18(3)1010IC5.83(6)17.89MC0(6)11511IC5.34(7)113.49MC0(7)13012DN0122DN012213IP5.14(6)113.71PR18(6)1014IC6.33(3)117.43MC0(3)135C(k+1)$199,880$199,000R(k+1,w)0.77340.7336*Thevaluein“Cost”columnistheallocatedrepaircostwiththefixedmaintenancecostinparentheses,wherecostsarein$1,000units.Symbolsdenotation:“DN”-DoNothing;IC-ImperfectCorrectiverepair;CR-CorrectiveReplacement;IP-ImperfectPreventiverepair;PR-PreventiveReplacement;MC-MinimalCorrectiverepair.2013book 53September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookSelectiveMaintenanceforComplexSystems39maintenancecostinScenario1isslightlyhigherthanScenario2($880or0.44%),theprobabilityofsuccessfullycompletingthemissioninScenario1remarkablyincreasesbynearly5.43%.Ifmaintenanceresourcesareunlim-ited,andeithercorrectiveorpreventivereplacementisperformedoneachcomponentbeforethenextmission(Scenario3),theprobabilityofmissionsuccessis0.8947withatotalmaintenancecostof$448,000.ItindicatesthemaintenancecostinScenario1decreasesby55.38%ascomparedwithScenario3,whiletheprobabilityofsuccessfullycompletingthemissioninScenario1decreasesbyonly13.56.3.2SelectiveMaintenanceforMSSwithMultistateComponentsTosolvetheselectivemaintenanceoptimizationproblemforanMSSwithmulti-statecomponents,thefollowingassumptionsareconsideredin[Pandeyetal.(2013b)]:1)Thesystemconsistsofmultiple,repairablecomponents.2)Thecomponentsaswellasthesystemmaybeinmultiplestates,i.e.componentsandsystemhasseveraldiscreteperformancelevels.3)Replacementbringsthecomponentbacktothebestpossiblestate.4)Maintenanceispossibleonlyduringmaintenancebreaks,nore-pair/maintenancecanbeperformedduringmissionsi.e.systemandcomponentsonlydegradeduringoperation.Maintenancemaybringacomponenttoabetterstate.5)Attheendofamission,thecurrentcomponent/systemstatesareobservable.6)SystemdegradationismodeledusingthehomogenousMarkovmodeli.e.transitiontimebetweencomponentsstatefollowexponentialdistribution.7)Limitedresources(budget,repairmanandtime)areavailableandtheamountofresourcesrequiredformaintenanceactivitiesareknownandfixed.Inplaceofbinarycomponents,[Pandeyetal.(2013b)]consideredmulti-statecomponentsforstudy.Basedonthestateofthecomponentsbeforemaintenance,systemdemandduringthenextmissionandavailablere-sources,themaintenancedecisionforeachcomponentinthesystemwasdetermined.InanMSS,eachcomponentinthesystemcanhavediscreteperformancerates(alsocalledstates)asshowninFig.7. 54September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book40ReliabilityModelingwithApplicationsComponent1ComponentiComponentnStatemnStatem1StatemiStatemn-1Statem1-1Statemi-1State1State1State1State0State0State0Fig.7AnMSSwithncomponentsIntheMSSshowninFig.7,therearenmultistatecomponentsandanycomponent‘i’(i=1,2...,n)has‘mi+1’states(0≤k≤mi).Thesojourntimeofacomponentinastateisindependentofthesojourntimeofanothercomponent,whichisinitsownstate.Here,state‘j=0’referstothecompletefailurestateofacomponent.Thisistheworststateforacomponent,whereas′j=m′isthebestpossiblestateforcomponentii[Pandeyetal.(2013b)].Anycomponenticanbein(mi+1)possiblestates(0≤k≤mi)andcorrespondingtoeachstatethereareassociatedperformanceratesgivenas:gi(t)={gi,0,gi,1,...,gi,mi},i=1,2,...,n(24)Theperformancerategi(t)ofacomponent‘i’atanyinstant(t≥0)isadiscreterandomvariableandcanhaveanyvaluefromgi,0togi,mi.TheperformancerateofanMSSisalsoarandomvariablethatdependsontheperformanceratesofthecomponents:G(t)=Ψ(g1(t),...,gn(t))(25)whereΨisthestructurefunction. 55September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookSelectiveMaintenanceforComplexSystems41ThesystemperformanceratecanalsohaveanydiscreteperformancevaluefromthesetG(t)={G0,G1,...,GS},whereG0,G1,...,GSaretheperformanceratesthatthesystemcanhaveatanygiventimedependingonthecomponentsperformancerateatthatinstant.Inthecaseofagroupofcomponentsconnectedinparallel,theperformancerateofthesubsystemisthesumoftheperformancesofthecomponents.Ifnscomponentsinasubsystemsareconnectedinparallelandatanytimeinstantt,theperformancesofthesecomponentsaregivenbygi(t)where1≤i≤ns,thentheperformanceofthissubsystemis:∑nSgsubsystem(t)=g(t)(26)ii=1Inthecaseofagroupofcomponentsconnectedinaseries,theperformanceofthesubsystemis:gsubsystem(t)=min(g(t))(27)i1≤i≤nSHence,foraseries-parallelsystematanyinstantt,,thesubsystem(thegroupofcomponentsconnectedinparallel)performanceisfirstdetermined,andthenthesystemperformanceiscalculated.Hence,theperformancelevelG(t)ofthesystemcanbeevaluatedas:subsystem∑nlgl(t)=(i=1gi(t),)l=1,2,...,ssubsystem(28)G(t)=ming(t)l1≤l≤sItisobviousfromtheabovediscussionthatatanyinstantt,thesys-temperformancecanbedescribedcompletelyifcomponentsperformancelevelsareknown.Ifthecomponentihasstateyibeforemaintenance,itisrequiredtofindthestatexiofeachcomponentaftermaintenancebreaksuchthatsystemperformanceduringthemissionissatisfied.Hence,xiisthedecisionvariablehere.ThefollowingmaintenanceoptionsarepossibleforthemultistatecomponentsofanMSS:1)Donothing:Noactionisperformed.Leavethecomponentasis.Inthisconditionthedecisionvariablexi=yi,i.e.thecomponentstatebeforeandafterthemaintenancebreakarethesame.2)Replacement:Theoldcomponentisreplacedandanewcomponentisinstalled.Afterreplacement,thecomponentstatebecomesxi=mi.3)Imperfectrepair/maintenance:Forimperfectrepair/maintenance,thestateofacomponentfromcompletefailure(yi=0)/non-failure(0 56September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book42ReliabilityModelingwithApplicationscaseyi 57September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookSelectiveMaintenanceforComplexSystems43MaintenanceDuration(T0)SystemPerformanceG(t)≥WG(t)PreviousMissionDemandLevel(W)MissionDuration(T)ChronologicalTime(t)Fig.8Systemperformanceduringamissionanddemandlevelwhereφisthedistributiveoperatordefinedbythefollowingequations:{(σ−W)P(T,x)ifσ≥WφP(T,x).z=(30)0ifσ 58September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book44ReliabilityModelingwithApplications3.2.2SelectivemaintenancemodelingUponthearrivalofthesystemformaintenance,thecomponents’perfor-mancestatesy=[y1,y2,...,yn]areknown.ThebudgetconstraintonthetotalmaintenancecostduringthemaintenancebreakisgivenbyC0andtheavailablemaintenancetimedurationislimitedtoT0.Thenon-linearformulationtomaximizetheprobabilityofsuccessfullycompletingthenextmissionisexpressedas:Objective:∑maxRS(W,T,x)=PJ(T,x)(33)GJ(t)≥WSubjectto:C≤C0(34)T≤T0(35)yi≤xi≤mi(36)Intheaboveformulation,CisthetotalcostofmaintenanceandTisthetimetoperformmaintenanceforthewholesystem(see[Pandeyetal.(2013b)]forfurtherdetails).Thisisatypicalconstrainednonlinearoptimizationprobleminvolvingintegervariablesonly.Anevolutionaryal-gorithmisusedtosolvethisoptimizationproblem.Differentialevolution(DE)isusedinthiscase.Tosolvetheproblemofselectivemaintenanceformultistatesystemswithmultistatecomponents,theexampleofcoaltrans-portationsystempresentedinSec.3.1isusedhere.However,itassumesthateverycomponentmaypossessmorethantwostates,ratherthanonlytwostatesasinSec.3.1.Thecapacityofeachcomponent,numberofcomponentstates,transi-tionprobabilities,andotherparametersarenotincludedhereduetospacelimitation.Allthesevaluescanbereadfrom[Pandeyetal.(2013b)].Weaimtohighlighttheadvantagesofimperfectrepair/maintenancepolicy;hence,wewilllimitourselvestodiscussingtheresultsrelatedtoimper-fectmaintenancerepaironly.Readersareencouragedtoread[Pandeyetal.(2013b)]forfurtherdetailsandanunderstandingoftheproblemandseeotherresultsrelatedtotheproblem. 59September17,201314:50Table6SelectiveMaintenancedecisionwithbothreplacementandimperfectmaintenance/repairasoptions(costsinthou-sandsof$andtimeinhrs)[Pandeyetal.(2013b)]Comp.(i)1234567891011121314SelectiveMaintenanceforComplexSystemsmi33322333334444yi00001121122101xi33021121322104BC:9023-ReliabilityModelingwithApplicationsActionsCRCRDNCRDNDNDNDNCRDNDNDNDNCRICi;x21.2016015.10000021.40000012.60iTi;xi2.251.7501.5500002.4000001.30Rs=0.91774,C=86.3,T=9.25xi22221222222121ActionsIRIRIRCRDNIMDNIMIMDNDNDNIRDNIICi;x16.201316.1015.1004.7506.136.400009.830iTi;xi1.751.451.751.5500.5500.630.900001.180Rs=0.9613,C=87.5096,T=9.7623*I=Onlyreplacementasoption,II=Imperfectmaintenanceincludedwithreplacementasoption.Ci;x=Thebud-igetusedinthemaintenanceofcomponenti,Ti;xi=Thetimeusedinthemaintenanceofcomponenti,DN=Donothing,IM=Imperfectmaintenance,IR=Imperfectrepair,CR=Componentreplacement.452013book 60September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book46ReliabilityModelingwithApplications3.2.3ResultsThecomparisonofresultswhenonlyreplacementispossibleandwhenimperfectrepair/maintenanceisalsoincludedarepresentedinTable6.Table6depictswhenonlyreplacementisconsideredasanoption,themaximumachievablesystemreliabilityis0.91774.Thetotalcostandtimeconsumedinthemaintenanceare86.30unitsand9.25hrs.,respectively.Inthisexample,unitcosts=$1000.Whileinthecaseofimperfectmainte-nance,themaximumachievablereliabilityincreasedto0.9613,anincreaseofabout5%.Intheimperfectmaintenance/repaircase,thecostandtimeusedduringmaintenanceare87.50unitsand9.76hrs,respectively.Itisclearthatbetterutilizationofresourcesispossibleifimperfectmaintenanceisincludedasanoptionforselectivemaintenanceratherthanusingreplacementastheonlymaintenanceoption.Ifonlyreplacementisoptedforasamaintenanceoptionthencomponents#1,2,4,9,and14areselected.Ifimperfectrepair/maintenanceisalsoincludedasamainte-nanceoptionthenonlycomponent#4isreplaced.Components#1,2,3,and13areselectedforimperfectrepairwhilecomponents#6,8,and9haveimperfectmaintenance.Nomaintenanceisperformedontheothercomponents.4ConclusionAneffectiveageandmaintenancecostbasedagereductionandhazardad-justmentfactorisdefinedin[Pandeyetal.(2013a)].Itisfoundthatforthesameinvestmentinrelativelyyoungercomponentsascomparedtooldercomponents,betterimprovementcanbeachieved.Amoregeneralizedhybridimperfectmaintenancemodelisusedtoformulatethecomponentimprovementafterrepair.Thisassumptionismorerealisticandmoregen-eral.Todefinetherelativeageofthecomponentacharacteristicconstant‘m’isalsoproposed,whichdetermineswhetheracomponentisrelativelyyoungerorolder.ItisfoundthatwhetherabinarysystemisconsideredoranMSS,imperfectmaintenance/repairisimportantinoptimizingselectivemaintenance.Underimperfectmaintenance/repairselectivemaintenanceoptimizationprovidesbetterreliabilitythanmaintenancedecisionswithreplacementonlyasanoption.Alloftheworksfocusedonsingleobjectiveonly,however,trade-offsbe-tweenmaintenancebudgets,time,andmissionsuccessprobability,aswellasotherresources(e.g.multiplerepairmen)needtobeaddressed.Multi-objectiveoptimizationapproachesmaybeusedtosolvethisproblem.For 61September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookSelectiveMaintenanceforComplexSystems47multistatesystemswithmultistatecomponentsaconstantfailureratewasused.ItwasassumedthatsojourntimefollowstheExponentialdistribu-tion,however,otherdistributions,e.g.theWeibulldistribution,canalsobeused.AcknowledgmentThisresearchwassupportedbytheNaturalSciencesandEngineeringResearchCouncilofCanada(NSERC).ReferencesBarlow,R.E.andHunter,L.E.(1960).Optimumpreventivemaintenancepoli-cies,OperationalResearch8,pp.90–100.Brest,J.,Greiner,S.,Boˇskovi´c,B.,Mernik,M.andZumer,V.(2006).Self-adaptingcontrolparametersindifferentialevolution:Acomparativestudyonnumericalbenchmarkproblems,IEEETransactionsonEvolutionaryComputation10,6,pp.646–657.Brown,M.andProschan,F.(1983).Imperfectrepair.JournalofAppliedProba-bility20,4,pp.851–859.Cassady,C.,Pohl,E.andMurdock,W.P.(2001a).Selectivemaintenancemodel-ingforindustrialsystems,JournalofQualityinMaintenanceEngineering7,2,pp.104–117.Cassady,C.R.,Murdock,W.P.andPohl,E.A.(2001b).Selectivemaintenanceforsupportequipmentinvolvingmultiplemaintenanceactions,EuropeanJournalofOperationalResearch129,2,pp.252–258.Chan,P.andDowns,T.(1978).Twocriteriaforpreventivemaintenance.IEEETransReliabR-27,4,pp.272–273.Chen,C.,Meng,M.Q.H.andZuo,M.J.(1999).Selectivemaintenanceoptimiza-tionformultistatesystems,inIEEECanadianConferenceonElectricalandComputerEngineering,pp.1477–1482.Dekker,R.andScarf,P.(1998).Ontheimpactofoptimisationmodelsinmain-tenancedecisionmaking:Thestateoftheart,ReliabilityEngineeringandSystemSafety60,2,pp.111–119.Dekker,R.,Wildeman,R.andVanDerDuynSchouten,F.(1997).Areviewofmulti-componentmaintenancemodelswitheconomicdependence,Mathe-maticalMethodsofOperationsResearch45,3,pp.411–435.Desai,A.andMital,A.(2006).Designformaintenance:Basicconceptsandreviewofliterature,InternationalJournalofProductDevelopment3,1,pp.77–121.Doyen,L.andGaudoin,O.(2004).Classesofimperfectrepairmodelsbasedonreductionoffailureintensityorvirtualage,ReliabilityEngineeringandSystemSafety84,1,pp.45–56.Iyoob,I.,Cassady,C.andPohl,E.(2006).Establishingmaintenanceresource 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64May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank 65September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookChapter3RandomReplacementPoliciesWonYoungYunandLiLiuDepartmentofIndustrialEngineering,PusanNationalUniversity,30-san,Jangjeon-dong,Geumjong-gu,Busan609-735,SouthKorea1IntroductionInmostliteraturerelatedtopreventivemaintenanceandreplacementpoli-cies,itisassumedthatwecanmaintainthesystempreventivelyatanytime.Butitissometimesimpracticaltoreplaceormaintainasysteminthestrictlypre-plannedtimebecauseitmayhavearandomworkcycleanditisimpossibleorimpracticaltoreplaceitinmid-cycle.Therandomagereplacementmodelisstudiedanalyticallyin[BarlowandProschan(1965)].[Nakagawa(2005)]summarizesrandommaintenancepoliciesinwhichthreereplacementcases(age,periodicandblockreplacement)arestudiedandarandominspectionmodelisalsoanalyzed.Forexample,aunitisreplacedattimeT,Y,oratfailure,whicheveroccursfirst,whereTisconstant(pre-planned)andYisarandomvariable(arandomagereplacementmodel).Secondly,asystemisreplacedatplannedtimeToratrandomtimeY,whicheveroccursfirst,andundergoesonlymini-malrepairatfailuresbetweenreplacements(randomperiodicreplacementmodel).[Nakagawaetal.(2011)]proposestworandomreplacementmod-elswithshortageandexcesscosts.[Chenetal.(2010)]studiesrandomreplacementpoliciesinwhichasystemisreplacedpreventivelyattimeTandnumberN.Forexample,aunitisreplacedattimeT,N,Y,oratfailure,whicheveroccursfirst,whereTandNaregivenandYisarandomvariable(randomagereplacementmodel).[ZhaoandNakagawa(2012)]considersoptimizationproblemsofreplacementfirstorlastinwhichtheconceptofreplacementlastisnewoneandthesystemisreplacedpreven-tivelyattimeTorY,whicheveroccurslast.[Zhaoetal.(2012)]studies51 66September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book52ReliabilityModelingwithApplicationsanagereplacementwithovertimepolicy.[Chen(2013)]proposesmodifiedmodelswithrandomageandperiodicreplacement.Randomreplacementmodelshavebeenalsostudiedasopportunisticreplacementpoliciesinwhichopportunistictimepointsofpreventivere-placementoccuraccordingtosomestochasticprocesses[DekkerandDijk-stra(1992)].Inthischapter,wesummarizerandomageandnumber-basedreplace-mentmodelsandconsiderrandomreplacementmodelswithdifferentcostterms.Tworandomreplacementmodelsarestudied:theplannedPR(Pre-ventiveReplacement)isassumedtobethemeanoftheactualPRtimeinthefirstmodelandtheopportunitytimesofPRareassumedtooccuraccordingtostochasticprocessesinthesecondmodel.Correctiveandpre-ventivereplacementcostsarealsoincludedinthemodelandadditionallyweconsiderabenefittermfromrandompreventivereplacementtimes.Largevarianceofthepreventivereplacementtimemeansthatwehavemoreflex-ibilityinmaintainingpreventivelythesystem.Innumericalexamples,weassumethatthesystemfailuredistributionisaWeibulldistributionandthepreventivemaintenancetimefollowsExponential,uniform,andtrian-gulardistributions.WedeterminetheoptimalmeanPRtimeminimizingtheexpectedcostratenumericallyandcomparetheoptimalPRtimesandtheexpectedcostratesintheexistingmodels.Thischapterisorganizedasfollows.InSection2,thestandardagereplacementandrandomreplacementmodelsaresummarizedandrandomagereplacementmodelsinwhichtheplannedPRtimeisthemeanoftheactualPRtimewithdifferentcosttermsarestudied.Time-basedandnumber-basedreplacementmodelsunderrandomopportunistictimesarereviewedinSection3.SomeextendedmodelsarediscussedinSection4.Finally,Section5concludesthechapter.Table1showsnotationinthischapter.2RandomAgeReplacementPoliciesInthischapter,westudypreventivereplacementpoliciesofasystemwithastochasticlifetimeunderthefollowingbasicassumptions.1.Thefailedsystemisreplacedbynewonecorrectively.2.Thesystemcanbereplacedpreventively.3.Theactualpreventivereplacementtimesarerandomvariables. 67September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookRandomReplacementPolicies53Table1NotationforrandomreplacementmodelsNotationc1Correctivereplacementcostc2PreventivereplacementcostcProfitratef(t),F(t),F(t),λ(t)Density,distribution,reliability,failureratefunctionsofsystemfailureµMeantimetosystemfailureTpPlannedPRtimeTaActualPRtime,arandomvariableg(·),G(·),G(·),r(·)Density,distribution,reliability,failureratefunctionsofactualpreventivemaintenancetime4.Thecorrectiveandpreventivereplacementtimesarenegligible.5.Thecorrectiveandpreventivereplacementcostsarec1andc2respec-tively(c1>c2).6.Thetimehorizonisinfiniteandthelongrunaveragecostisusedastheoptimizationcriterion.Thus,wewanttofindtheoptimalpreventivereplacementpoliciestominimizetheexpectedcostrate.2.1StandardAgeReplacementPolicyInthissubsection,weconsidertheagereplacementpolicyinwhichthesys-temisplannedtobereplacedpreventivelyatageTp.Thus,ifthesystemisfailedbeforetheplannedTp,thesystemisreplacedbynewonecorrectively.Otherwise,thesystemisreplacedpreventively.Firstly,iftheactualpreventivetimescanbeequaltotheplannedpre-ventivetimes,theexpectedcostrateisgiven[Nakagawa(2005)](c1−c2)F(Tp)+c2C1(Tp)=∫T.(1)pF(t)dt0FromdC1(Tp)/dTp=0,theoptimalTpwegetassolutionoftheequation∫Tpc2λ(Tp)F(t)dt−F(Tp)=.(2)0c1−c2Supposethatthefailurerateiscontinuousandstrictlyincreasing.Ifλ(∞)>c/[µ(c−c)],thenthereexistsafiniteanduniqueT∗thatsatisfies112pEquation(2).Otherwise,theoptimalT∗=∞.p 68September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book54ReliabilityModelingwithApplications2.2RandomAgeReplacementPolicyInthissubsection,weassumethattheactualpreventivereplacementtimesarerandomvariablesdependentontheplannedPRtime.IftheplannedpreventivereplacementtimeisTp,theactualPRtimes,TaarerandomvariablesofwhichthedistributionisdependentonTp.Inthissection,weassumethatthemeanvalueoftheactualPRtimesistheplannedPRtime,E[Ta]=Tp.Thus,thesystemisreplacedattimeTa,oratfailure,whicheveroccursfirst.ButtheactualPMtimeTaisarandomvariableandtheexpectedcostrateisgiven[Nagakawa(2005)],∫∞(c1−c2)0G(t;Tp)dF(t)+c2C2(Tp)=∫∞,(3)G(t;Tp)F(t)dt0whereG(t;Tp)isthereliabilityfunctionoftheactualPRtimeinwhichthemeanvalueisTp.Theoretically,therandomagereplacementpolicyisworsethanthestandardagereplacementpolicy(deterministicPM)[BarlowandPoschan(1965)].Thus,minC1(Tp)≤minC2(Tp).TpTpTherefore,ifwecanmaintainpreventivelyonplannedPMtime,weshouldkeeptheplannedPRtimes.Butinpractice,itissometimesimpos-sibleornoteconomicaltoreplaceaniteminthepreplannedtime.Some-times,ifwemaintainearlyordelaythePRtimes,wecansavemuchtheproductioncostinducedbyinterruptingcontinuousworks.Inrandomre-placementmodels,themeanoftheactualPRtimeistheplannedPRtimeanditsvarianceisrelatedtoflexibilityofPR.WecangetsomeprofitsfromchangingPRtimeanditisassumedthattheprofitisproportionaltothestandarddeviationoftheactualPRtime.Thus,theexpectedcostratewithaprofittermisgiven∫∞(c1−c2)0G(t;Tp)dF(t)+c2−c·SD(Ta)C3(Tp)=∫∞.(4)G(t;Tp)F(t)dt0ItisdifficulttocomparethethreeexpectedcostratesanalyticallyandweinvestigatethethreeexpectedcostratesandtheoptimalPRtimesnumericallyincasethatthesystemfailuredistributionisaWeibulldistri-butionwithscaleparameter,λ=1andc2=1. 69September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookRandomReplacementPolicies551.ExponentialdistributioncaseIftheactualpreventivereplacementtimefollowsanExponentialdistribu-tionofwhichthemeanistheplannedPRtime,Tp,theexpectedcostratewiththeprofittermis∫∞(c−c)αtα−1e−t/Tp−tdt+c−cT120∫2pC3(Tp)=∞e−t/Tp−tdt.(5)0TheoptimalPRtimeTpminimizingtheexpectedcostrateisobtainednumerically.Fornumericalexamples,c1=4andc2=1.Firstly,weobtaintheexpectedcostratewithouttheprofitratetermandfromFig.1,weknowtheoptimalPRintervaldecreasesastheshapeparameterofWeibulldistributionincreases.Secondly,weconsidertheexpectedcostratewiththeprofittermandTable2givestheoptimalT∗andexpectedcostrateC(T∗)fordifferentppshapeparameterαandcostratioc/(c−c).ItindicatesthatT∗increases212pwithratioc/(c−c),C(T∗)decreaseswithαandratioc/(c−c).For212p212largec2/(c1−c2),wedonotneedpreventivereplacement.Fig.1OptimalTpandexpectedcostrateswithoutprofitterm(Exponential,c1=4) 70September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book56ReliabilityModelingwithApplicationsTable2OptimaltimeTpanditscostrateC(Tp)withc=1(randomagereplacementpolicywithExponentialdistributioncase)c2α=2.0α=2.5α=3.0α=5.0c1−c2TpC(Tp)TpC(Tp)TpC(Tp)TpC(Tp)0.010.149727.58600.085619.84300.971815.76500.125010.40000.020.163420.51300.116014.93000.126112.38700.15108.83030.050.191012.35800.180310.14300.18568.88550.20316.95690.100.28898.55560.26867.46310.26476.79020.27075.66570.200.54115.82930.45405.37810.42375.06480.40004.47400.504.95253.37611.91693.32391.42623.25641.04203.07011.00∞2.2568∞2.2540∞2.2396∞2.17822.00∞1.6926∞1.6905∞1.6797∞1.63365.00∞1.3541∞1.3524∞1.3438∞1.30692.UniformdistributioncaseNowweconsideradifferentdistributioncaseinwhichtheactualPRtimefollowsauniformdistributionwithmeanTpandanallowedlimitδ.Then,thedensityofTaisgivenby1g(t;Tp)=,Tp−δ≤t≤Tp+δ.2δThus,theexpectedcostrateis∫Tp+δ1/(2δ)Tp−δ[(c1+c2)δ+(c1−c2)(√Tp−t)]f(t)dt+c1F(Tp−δ)+c2F(Tp+δ)−cδ/3CU(T)=∫Tp+δ22.(6)1/(4δ)Tp−δ[2t(Tp+δ)−t−(Tp−δ)]f(t)dt∫Tp−δ+0tf(t)dt+TpF(Tp+δ)WeconsidernumericalexamplesandthescaleandshapeparametersoftheWeibulldistributionaregivenas1and2.Figures2–4showtheexpectedcostratefunctionforc1=4andc=0.5,1.Innumericalresults,wecanfindtheoptimalδminimizingtheexpectedcostrate.Additionally,whenδincreases,theexpectedcostratedecreasesandincreasesinFigs.2–4.Thus,eventhoughitisnoteasytofindtheoptimal(δ,T)minimizingtheexpectedcostrateanalytically,wemayfindnearoptimalsolutionsbynumericalmethodsinmostcases.3.TriangulardistributioncaseInthissubsection,weassumethattheactualpreventivereplacementtimefollowsatriangulardistributionwithmeanTpandanallowedlimitδ.Thus, 71September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookRandomReplacementPolicies57Fig.2Expectedcostrateswithoutprofitterm(c1=4):uniformdistributioncaseFig.3Expectedcostrateswithdifferentallowedlimits(c1=4,c=0.5):uniformdistributioncasethedensityofTaisgivenby{x−Tp+δδ2,Tp−δ≤t≤Tpg(t;Tp)=Tp−x+δ(7)δ2,Tp≤t≤Tp+δThus,theexpectedcostrateisobtainedfromEquation(4)asintheuniformcase.Fromthenumericalexamples,wegettheoptimalpreventive 72September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book58ReliabilityModelingwithApplicationsFig.4Expectedcostrateswithdifferentallowedlimits(c1=4,c=1):uniformdistri-butioncaseFig.5Expectedcostrateswithoutprofit(c1=4):triangulardistributioncasereplacementtimeT∗whichminimizestheexpectedcostratewhenthepscaleandshapeparametersofWeibulldistributionaregivenby1and2respectivelyandPRcostis1.Figures5–7showthatwhenδincreases,theexpectedcostratedecreases. 73September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookRandomReplacementPolicies59Fig.6Expectedcostrateswithdifferentallowedlimits(c1=4,c=0.5):triangulardistributioncaseFig.7Expectedcostrateswithdifferentallowedlimits(c1=4,c=1):triangulardistributioncase2.3NumericalExamplesInthissection,westudymorenumericalexamples,T∗andassumethatpthesystemfailurefollowsaWeibulldistributionandthescaleandshapeparametersaregivenby1and2respectively.Firstly,weconsiderthe 74September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book60ReliabilityModelingwithApplicationsTable3OptimaltimeTpanditscostrateC(Tp)(randomagereplacementpolicywithuniformdistributioncase)c2δ=0.1δ=0.2δ=0.3δ=0.5c1−c2TpC(Tp)TpC(Tp)TpC(Tp)TpC(Tp)0.010.116123.07250.155630.30150.207839.13030.337457.03920.020.153815.27630.186318.14250.233021.93770.356530.13480.050.23319.26260.257810.04000.296411.16220.408713.82450.100.32596.47400.34656.74770.38027.15930.48428.19510.200.46104.57960.47974.67030.51074.80980.61045.17290.500.74442.95780.76392.97550.79673.00270.90273.07331.001.09872.18291.12252.18651.16212.19201.28602.20542.001.69991.68871.73331.68901.78711.68941.94191.69025.003.40711.35413.46681.35413.55041.35413.74341.3541Table4OptimaltimeTpanditscostrateC(Tp)(randomagereplacementpolicywithtriangulardistributioncase)c2δ=0.1δ=0.2δ=0.3δ=0.5c1−c2TpC(Tp)TpC(Tp)TpC(Tp)TpC(Tp)0.010.108421.60240.130425.61370.161230.75420.238041.97860.020.147814.73290.165116.28100.191118.39150.261023.25460.050.22899.12360.24159.53110.261610.14680.320811.62400.100.32246.42620.33286.56740.34986.78990.40217.37480.200.45794.56390.46724.61040.48264.68460.53144.89770.500.74112.95470.75092.96380.76712.97840.81893.02051.001.09482.18221.10662.18411.12642.18721.18962.19572.001.69431.68861.71121.68881.73931.68901.82701.68975.003.39631.35413.42911.35413.48101.35413.62791.3541expectedcostrateswithoutprofitterm.Tables3and4showtheoptimumT∗andexpectedcostrateC(T∗)fortheallowedlimitδandcostratioppc2/(c1−c2).Thestandardagereplacementcases(δ=0)givetheminimumcost.ThetwotablesindicatethatT∗increaseswithδandc/(c−c),andp212C(T∗)increaseswithδbutdecreaseswithc/(c−c).Incasethatthep212allowedlimitδandcostratioc2/(c1−c2)aresame,triangulardistributioncasesgivelesscoststhanuniformcases.Secondly,weconsidertheexpectedcostratewithprofitterm.Tables5and6presenttheoptimalδ∗andT∗andminimalexpectedcostrateC∗pfordifferentvaluesofc2/(c1−c2)inuniformandtriangulardistributioncases,respectively(c=1,2).ItindicatesthatT∗andC∗decreasewithpc2/(c1−c2).Fromthetables,wecanfindthattherandomagereplacementpolicywithprofittermobtainedbyflexiblePRtimescanbebetterthanthedeterministicagereplacementpolicy(standardagereplacementpolicy). 75September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookRandomReplacementPolicies61Table5OptimalδandTpwithc2=1andc=1(randomagereplacementpolicywithprofitterm)c2UniformdistributioncaseTriangulardistributioncasec1−c2δTpC(Tp,δ)δTpC(Tp,δ)0.010.010.100019.99190.010.100019.99230.020.020.141314.13010.030.141414.13090.050.050.22348.92250.060.22338.92230.100.100.31656.28750.140.31646.28730.200.220.45454.40030.310.45424.39980.501.301.30192.25940.790.79042.60081.001.551.55141.13011.211.21041.60872.001.641.64870.56971.731.73690.87525.001.691.71750.23272.162.16160.3538Table6OptimalδandTpwithc2=1andc=2(randomagereplacementpolicywithprofitterm)c2UniformdistributioncaseTriangulardistributioncasec1−c2δTpC(Tp,δ)δTpC(Tp,δ)0.010.020.099619.91730.020.099619.91890.020.040.140414.02250.050.140314.02000.050.090.21978.74050.130.21978.74020.100.190.30776.00810.280.30826.00680.200.440.44563.91520.420.42333.99440.500.620.62471.93970.620.62392.20531.000.730.73581.04330.800.80321.31542.000.790.81390.54840.960.97220.73695.000.830.89050.23251.101.13460.32023PreventiveReplacementPolicieswithRandomOpportunityTimesInthissection,weassumethatpreventivereplacementsareallowedatop-portunitytimeswithPRcost,c2andtheopportunitytimesoccuraccordingtoarenewalprocesswithadistributionH(t)inwhichtherenewalfunctionisM(t).Therenewalprocessofopportunitiesisassumedtoberesetaftercorrectiveandpreventivereplacement.3.1OpportunityAgeReplacementModelWeconsideranagereplacementpolicyinwhichthesystemisreplacedpreventivelyatthefirstopportunitytimeafterTp.Theagereplacementpolicyisstudiedby[DekkerandDijkstra(1992),Chenetal.(2010)andZhaoetal.(2012)]. 76September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book62ReliabilityModelingwithApplicationsFirstly,wederivetheresiduallifeintherenewalprocessofopportunities.Lettheresiduallifebeγ(Tp)andD(t;Tp)bethedistributionofγ(Tp).Thenthedistributionoftheresiduallifeisgiven[Nakagawa(2011)]∫TpP(γ(Tp)≤t)=D(t;Tp)=H(Tp+t)−H(Tp+t−u)dM(u).(8)0ThentheactualPRtimeisarandomvariableandTa=Tp+γ(Tp).Theexpectedcostofonecycleisc1P(X≤Ta)+c2P(X>Ta)=c1−(c1−c2)P(X>Ta)∫∞=[−(c1−c2)P(X>Tp+t)+c1]dD(t)0∫∞=c1−(c1−c2)F(Tp+t)dD(t;Tp).0Theexpecteddurationofonecycleis∫∞E[min(X,Tp)]=E[min(X,Tp+t)]dD(t;Tp)0∫∞∫t+Tp=F(x)dxdD(t;Tp)00∫Tp∫∞=F(x)dx+F(x+Tp)D(x)dx.00Theexpectedcostrateisgivenas∫∞c1−(c1−c2)0F(Tp+t)dD(t;Tp)C4(Tp)=∫Tp∫∞,(9)0F(x)dx+0F(x+Tp)D(t;Tp)dtwhereD(·)isthedistributionfunctionoftheresiduallifetime.Thismodelissimilarwiththeagereplacementwithovertimepolicyintroducedby[Zhaoetal.(2012)].3.2Number-basedReplacementPolicyInthissection,weconsiderapreventivereplacementpolicybasedonthenumberofopportunities.ThesystemisreplacedpreventivelybeforefailureattheNthopportunitytime.TheactualPRtimeisarandomvariableandtheNthopportunitytime.ThedistributionfunctionofNthopportunity 77September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookRandomReplacementPolicies63timeisthej−foldStieltjesconvolution,H(N)(t)ofH(t)withitself.c1P(X≤Ta)+c2P(X>Ta)=c1−(c1−c2)P(X>Ta)∫∞=[−(c−c)P(X≤t)+c]dH(N)(t)1210∫∞=c−(c−c)F(t)dH(N)(t).1120Theexpecteddurationofacycleis∫∞∫∞∫tE[min(X,T)]=E[min(X,t)]dH(N)(t)=F(x)dxdH(N)(t)a000∫∞=F(x)[1−H(N)(t)]dt.0Thentheexpectedcostrate(Chen,2013)is∫∞c−(c−c)F(t)dH(N)(t)1120C5(N)=∫∞.(10)F(x)[1−H(N)(t)]dt03.3NumericalExamplesWeconsidernumericalexamplesinwhichthesystemfailuredistributionisaWeibulldistributioninwhichthescaleandshapeparametersaregivenby1and2.APoissonprocessisassumedandinter-arrivaltimesofopportunitiesfollowsanExponentialdistributionwithafinitemean1/θ.Then,D(t;T)=1−e−θtandH(N)(t)=1−∑N−1(θt)j/j!e−θt,thepj=0expectedcostratein(9)is∫∞2c−(c−c)θe−θt−(t+Tp)dt1120CD(Tp)=∫Tp−t2∫∞−θt−(t+T)2,(11)edt+epdt00andtheexpectedcostratein(10)isc−(c−c)∫∞e−t2d[1−∑N−1(θt)j/j!e−θt]1120j=0CO(N)=∫∞∑N−1j−θt−t2.(12)(θt)/j!edt0j=0Table7presentstheoptimalT∗andexpectedcostrateC(T∗)fordif-ppferentc/(c−c)and1/θ.ItindicatesthatT∗increaseswithc/(c−c)212p212andθ,andC(T∗)decreaseswithc/(c−c)andθ.Table8presentsthep212optimalN∗andexpectedcostrateC(N∗)fordifferentc/(c−c)and2121/θ.ItindicatesthatN∗increaseswithc/(c−c)andθ,andC(T∗)de-212creaseswithc2/(c1−c2)andθ.Additionally,age-basedreplacementpolicy 78September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book64ReliabilityModelingwithApplicationsTable7OptimaltimeTpanditscostrateC(Tp)(randomagereplacementpolicywithexponentiallydistributedopportunitytime)c21/θ=0.11/θ=0.21/θ=0.31/θ=0.5c1−c2TpC(Tp)TpC(Tp)TpC(Tp)TpC(Tp)0.010.042827.67150.026040.18600.019154.29150.015366.49600.020.075517.04710.054322.35800.037228.76450.030134.51000.050.14939.72230.116611.24300.086913.28590.073015.24200.100.23926.62950.19327.16190.15867.93780.13688.71890.200.37274.62850.31434.79960.27855.05800.24895.33060.500.65582.96640.63022.98940.55543.04200.52143.09091.001.01122.18451.00622.13930.91782.19780.88272.20612.001.61481.68881.57661.68921.53891.68961.50861.69015.003.32591.35413.31091.35413.19661.35413.14601.3541Table8OptimaltimeNanditscostrateC(N)(Number-basedreplacementpolicywithexponentiallydistributedopportunitytime)c21/θ=0.11/θ=0.21/θ=0.31/θ=0.5c1−c2NC(N)NC(N)NC(N)NC(N)0.01129.4627140.6208154.3889166.53570.02219.3927122.9867128.9488134.58730.05210.8717112.4063113.6848115.41830.1037.256827.862218.596819.02860.2054.920835.200925.447425.76220.5083.061853.128833.192333.23641.00132.211282.226262.237352.24452.00221.6914141.6921101.692491.69255.00321.3541191.3541141.3541111.3541outperformsnumber-basedpolicybutthegapbetweentwopoliciesintheexpectedcostrateisnotbig.4ExtendedModelsInthissection,weconsidertwoextendedmodels:Case1:Inthiscaseweconsidersomeextendedmodels.InSection3,weassumearenewalprocessforopportunitytimes.First,weassumethattheopportunitytimesoccuraccordingtoanon-homogeneousPoissonprocess∫t(NHPP)withintensityfunctionλ(t)andletΛ(t)=λ(x)dx.0Then,theresiduallifeafterTphasthefollowingdistributionfunctionP(γ(Tp)≤t)=D(Tp,t)=1−P{N(T+t)−N(T)=0}=1−e−(Λ(Tp+t)−Λ(Tp)).(13)pp 79September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookRandomReplacementPolicies65Thus,theexpectedcostrateunderrandomagereplacementpolicycanbeobtainedfromEquation(9).ForNHPPcase,n∑−1[Λ(t)]ke−Λ(t)H(N)(t)=P(S≤t)1−.nk!k=0Thus,theexpectedcostrateofnumber-basedreplacementpolicycanbeobtainedeasilyfromEquation(10).Case2:wehavetwotypesofpreventivereplacement(randomanddeter-ministic)withdifferentPRcosts.Thenweconsideranewagereplacementpolicywithtwoparametersinwhichthefailedsystemisreplacedcorrec-tivelybeforePRtimes.IftherandomPRtimeTrislessthanT2andgreaterthanT1,T1 80September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book66ReliabilityModelingwithApplicationsstandardagereplacementpolicy.Time-basedandnumber-basedreplace-mentmodelsunderrandomopportunistictimesarereviewedandsomeex-tendedmodelsarediscussed.Inextendedmodels,wederivedtheexpectedcostratesbutmoreanalysisisrequiredforfurtherstudies.AcknowledgementThisresearchwassupportedbyBasicScienceResearchProgramthroughtheNationalResearchFoundationofKorea(NRF)fundedbytheMinistryofEducation,ScienceandTechnology(No.2010-0025084).ThefirstauthorisverygratefultoProfessorToshioNakagawawhostimulatedhisresearchinterestinmaintenancemodelsandpoliciesthroughmanypapersandfourbooksinreliabilityandmaintenance.ReferencesBarlow,R.E.andProschan,F.(1965).MathematicalTheoryofReliability(Wiley,NewYork).Chen,M.,Mizutani,S.andNakagawa,T.(2010).Randomandagereplacementpolicies,InternationalJournalofReliability,QualityandSafetyEngineer-ing17,pp.27–39.Chen,M.,Nakamura,S.andNakagawa,T.(2010).Replacementandpreventivemaintenancemodelswithrandomworkingtimes,IEICETrans.Fundamen-talsE93-A,pp.500–507.Chen,M.(2013).Optimalrandomreplacementmodelswithcontinuouslypro-cessingjobs,AppliedStochasticModelsinBusinessandIndustry29,pp.118–126.Dekker,R.andDijkstra,M.C.(1992)Opportunity-basedagereplacement:Ex-ponentiallydistributiontimesbetweenopportunities,NavalResearchLo-gistics,39,pp.175–190.Nakagawa,T.(2005).MaintenanceTheoryofReliability(Springer,London).Nakagawa,T.(2011).StochasticProcesswithApplicationstoReliabilityTheory(Springer,London).Nakagawa,T.,Zhao,X.andYun,W.Y.(2011).Optimalagereplacementandin-spectionpolicieswithrandomfailureandreplacementtimes,InternationalJournalofReliability,QualityandSafetyEngineering,18,pp.405–416.Zhao,X.andNakagawa,T.(2012).Optimizationproblemsofreplacementfirstorlastinreliabilitytheory,EuropeanJournalofOperationalResearch223,pp.141–149.Zhao,X.,Qian,C.andNakagawa,T.(2012).Agereplacementwithovertimepolicy,inThe5thAsia-PacificInternationalSymposiumonAdvancedRe-liabilityandMaintenanceModelingV(Nanjing,China),pp.661–668. 81September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookChapter4OptimalReplacementIntervalofaDualSystemSatoshiMizutaniDepartmentofMediaInformations,AichiUniversityofTechnology,50-2Manori,Nishihazama-cho,Gamagori443-0047,Japan1IntroductionInthischapter,weconsideroptimalreplacementpoliciesinwhichasystemoperatesasadualsystemwhenitisreplaced.Thesystemusuallyworksbyaagingunitsystemwhichisinrandomfailureperiodorwearoutfailureperiod.Whenitisreplacedtoafreshunitwhichisininitialfailureperiod,thesystemworksasadualsystemforawhile.Thatis,inordertoavoidlosscostoftheinitialandwearoutfailure,thesystemworksoperatesasadualsystemfromintroducingthefreshunittostoppingtheagingunit.Inrecentyears,informationsystemwithnetworkhasbeengreatlyde-velopedandbecomewidelyused,andhencefailuresofthesystemcausechangingofthedesignandasocialconfusion.Forexample,whenasystemnewlybeginstooperate,sometroublessuchassoftwarefailureorconfu-sionofusingthenewcontrolinterfacemightbeoccurs.Ontheotherhand,wedelayatimeofreplacement,thenriskofoccurrenceofwearoutfailurebecomelarge.Weshoulddeviseacountermeasure,andproposethatthesystemoperatesasadualsystemforawhilewhenthesystemisreplaced.However,thecostofoperatingasadualsystemwillincurmorethantheoneofoperatingasasinglesystem.Accordingly,itisgreatlyimportantandindispensablynecessarytoplansuitablymaintenancefromtheviewpointsofreliabilityandeconomics.Therehavebeenmanystudiesofmaintenanceandreplacementpoli-ciesusingreliabilitytheory[Barlow(1965);Osaki(1992);Nakagawa67 82September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book68ReliabilityModelingwithApplications(2011)].Periodicreplacementwithminimalrepairwasconsideredin[Bar-low(1965)].Periodicreplacementwithminimalrepairissummarizedin[Nakagawa(1981)].Theperiodicreplacementmodelwithtwounitiscon-sideredin[Nakagawa(1987)].Inthemodel,whenunit-1fails,itundergoesminimalrepair,andwhenunit-2fails,thesystemisreplacedwithoutrepair-ing.Minimalrepairmodelinwhichthefailureoccursatanon-homogeneousPoissonprocessaredescribedin[Murthy(1991)].Therehavebeenalsomanystudiesofmaintenancepoliciesformulti-unitsystems.Nakagawaconsideredanalyticallyoptimalpolicytodecidethenumberofunits[Nak-agawa(1984)].Inthemodels,thesystemfailswhenallunitshavefailed.Eachunitfailsfromshocks,independentlyoftheotherunits.MurthyandNguyenproposedthemodelthattheunitsfailwithinteraction[Murthy,Nguyen(1985a,b)].YasuiandNakagawasummarizedoptimumpoliciesforaparallelsystem[Yasui,Nakagawa,Osaki(1988)].Mizutani,KoikeandNakagawaconsideredthereplacementpolicythatthesystemoperatesasadualsystemfromthebeginningoffreshunittothestoppingofagingunit.Inthemodel,theagingunitisinwearoutfailureperiod[Mizutani(2010)].Weconsiderasystemwhichconsistsofoneunit,andthesystemhavebeenoperatedalreadybeforetime0,andwecalltheunitagingunit.Inthispaper,weextendthemodels[Mizutani(2010)]byintroducingtwocasesthattheagingunitisinrandomfailureperiodorwearoutfailureperiod.Wereplacetheagingunitwithnewonewhichiscalledfreshunit.Then,weproposethatthesystemoperatesasadualsystemfromthebeginningoffreshunittothestoppingofagingunit.Especially,wetreattwocases:(i)agingunitisinrandomfailureperiod,and(ii)inwearoutfailureperiod.Wederiveanalyticallyoptimaltimesofthestoppingofagingunit,whichminimizetheexpectedcost.Further,forthemodelthattheagingunitisinwearoutfailureperiod,wederiveoptimaltimeoftheintroducingofthefreshunit.Further,weconsideraproblemtoobtainoptimaltimesetwhichconsistsofatimeofstoppingagingunitandoneofintroducngfreshunit.NumericalexamplesaregivenwhenthefailuredistributionsofthefreshunitisaWeibulldistribution,andtheagingunitisanexponentialdistributionorWeibulldistribution.2ModelingWemaketwomodels:(1)thesystemisreplacedwhentheagingunitoperateinrandomfailureperiod,(2)thesystemisreplacedwhenagingunit 83September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookOptimalReplacementIntervalofaDualSystem69operateinwearoutfailureperiod.Firstly,weshowcommonassumptionfortwomodels:(1)FreshunithasafailuredistributionF(t)=1−e−Hf(t),fandthefailurerateh(t)≡H′(t)isamonotonicdecreasingfunction,ff∫twhereH′(t)isadifferentialfunctionofH(t),i.e.,H(t)=h(u)du.fff0f(2)Whenthesystemfails,aminimalrepairisdone:Thatis,thefailurerateremainsundisturbedaftertherepair,i.e.,thesystemaftertherepairhasthesamefailurerateasbeforethefailure[Nakagawa(2005)].Itisassumedthattherepairtimesarenegligible.(3)Costcaisaminimalrepaircostwhentheagingunitoperatesasadualsystem,andαaisanadditionalrepaircostwhenitoperatesasasinglesystem.Costcf(cf≥ca>0)isaminimalrepaircostwhenthefreshunitoperatesasadualsystem,andαfisanadditionalrepaircostwhenthefreshunitoperatesasasinglesystem.Furthermore,costcdistheoperatingcostperunitoftime,andcristhereplacementcost.(1)ReplacementinRandomFailurePeriodWeconsideranagingunitwhichisreplacedattimeta(>0)whenitisinarandomfailureperiodandhasanexponentialdistribution(1−e−λt).Inthiscase,thismodelisacasethatwecanpredictthetimeatwhichthestateoftheagingunitchangetoawearoutfailureperiod,andcanreplaceitbeforeawearoutfailureperiod.Wecanadoptthismodeltoanothercasethatthesystemisreplacedforareasonbesidethesystemwearout.Forexamples,wecanconsiderthereasonssuchasatechnologicalinnovation.Weassumethefollowingassumptionsforthismodel:(1)Thefreshunitisanintroducedattime0,andisininitialfailureperiodfromtime0toT(T>0).(2)Thestoppingtimeofanagingunitista(0≤ta≤T).(3)TheagingunithasanexponentialdistributionF(t)=1−e−λtawithfinitemean1/λ.Theexpectednumberoffailureofthefreshunitfrom0totimeTisHf(T).Especially,whenthefreshunitoperatesasasingleunitfromtimeta 84September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book70ReliabilityModelingwithApplicationsfailurerateofagingunitminimalrepaircafailurerateoffreshunitcfcfcfcf+αfcd0taTtimeFig.1ReplacementmodelinrandomfailureperiodtotimeT,theexpectednumberoffailureofthefreshunitisHf(T)−Hf(ta).Theexpectednumberoffailureofagingunitfrom0totimetaisλta.Thus,theexpectedcostC1(ta)fromtime0totimeTisgivenbyC1(ta)=cfHf(T)+αf[Hf(T)−Hf(ta)]+(caλ+cd)ta+cr.(1)(2)ReplacementinWearoutFailurePeriodWeconsideranagingunitwhichisreplacedinawearoutfailureperiod(Fig.2).Wemakefollowingassumptions:(1)Theagingunitisinawearoutfailureperiodattime0,andthefreshunitisintroducedattimetf.Thefreshunitisinaninitialfailureperiodfromtftotf+T,anditsstatechangestorandomfailureperiod.Theagingunitisstoppedattimeta(tf≤ta≤tf+T).(2)TheagingunithasafailuredistributionF(t)=1−e−Ha(t),andtheafailurerateh(t)≡H′(t)isamonotonicincreasingfunction.aa 85September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookOptimalReplacementIntervalofaDualSystem71failurerateofagingunitcaminimalrepair+αacacafailurerateoffreshunitcfcfcfcf+αfcd0tftatf+TtimeFig.2ReplacementinwearoutfailureperiodThefailuredistributionF(t)ofaduplexsystemattimet(0≤t≤T)is1−e−Hf(t)t 86September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book72ReliabilityModelingwithApplicationsTherefore,thefailurerateattimet(tf≤t≤ta)r(tf,ta)isf(t)r(tf,ta)==hf(t)+ha(t−tf).(2)1−F(t)Theexpectednumberofthefailureofthefreshunitfromtatotimetf+TisHf(T)−Hf(ta).Thus,theexpectedcostrateC2(ta,tf)from0totf+TisgivenbycaHa(ta)+αaHa(tf)+cfHf(T)+αf[Hf(T)−Hf(ta−tf)]+cd(ta−tf)+crC2(ta,tf)=.(3)tf+TTheequation(1)istheexpectedcostfromtime0totimeT,andisnottheexpectedcostrate,becausetheintervaltimeTisgivenasaconstantvalue.However,theequation(3)istheexpectedcosrate,becausetheintervaltimetf+Tisnotgivenasaconstantvalue.3OptimalPolicies3.1ReplacementinRandomFailurePeriodWeseekanoptimalt∗(0≤t∗≤T)whichminimizestheexpectedcostrateaaC1(ta).DifferentiatingC1(ta)withrespecttotaandsettingitequalto0,λca+cdhf(ta)=.(4)αfBecausehf(ta)isamonotonicdecreasingfunction,wehavethefollowingoptimalpolicy:(i)If(λc+c)/α≥h(0)thent∗=0adffa(ii)Ifhf(0)>(λca+cd)/αf>hf(T)thenthereexistsafiniteanduniquet∗(0 87September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookOptimalReplacementIntervalofaDualSystem73LettingL2a(ta)betheleft-handofaboveequation.Becausehf(t)(t>0)isamonotonicincreasingfunctionandha(t)(t>0)isamonotonicdecreasingfunction,L2a(t)(t>0)isamonotonicincreasingfunction.Therefore,wehavefollowingoptimalpolicy:(i)Ifc≥L(t)thent∗=0d2afa(ii)IfL(t)>c>L(t+T)thenthereexitsafiniteanduniquet∗2afd2afawhichsatisfies(5).(iii)IfL(t+T)≥cthent∗=t+T2afdaf3.2.2AnalysisofOptimaltfNext,weseekoptimalt∗thatminimizetheexpectedcostrateC(t,t).f2afDifferentiatingC2(ta,tf)withtfandsettingitequalto0,αa[ha(tf)(tf+T)−Ha(tf)]+αf[hf(ta−tf)(tf+T)+Hf(ta−tf)=caHa(ta)+(cf+αf)Hf(T)+cd(ta+T)+cr.(6)LettingL2f(tf)betheleft-handofaboveequation.BecauseL′(t)=αh′(t)(t+T)−αh′(t−t)>0,2ffaaffffafL2f(tf)isamonotonicincreasingfunction.Therefore,wehavefollowingoptimalpolicy:(i)IfL(0)≥cH(t)+(c+α)H(T)+c(t+T)+cthent∗=02faaafffdarf(ii)IfL2f(ta)>caHa(ta)+(cf+αf)Hf(T)+cd(ta+T)+cr>L2f(0)thenthereexistsafiniteanduniquet∗whichsatisfies(6).f(iii)IfcH(t)+(c+α)H(T)+c(t+T)+c≥L(t)thent∗=taaafffdar2fafa3.2.3AnalysisofOptimaltaandtfWeseekoptimalset(t∗,t∗)thatminimizetheexpectedcostrateC(t,t).af2afForthepurpose,wecanusefollowingwell-knowntheoremas”SecondDerivativeTest”:Supposethatz=f(x,y)hascontinuousfirstandsecondpartialderivatives,and∂z/∂x=0and∂z/∂y=0.Let∆≡∂2z/∂x∂y−∂2z/∂x∂x·∂2z/∂y∂y<0.If∆>0and∂2z/∂x∂x<0,then(x,y)isapontthatzisalocalmaximum.If∆>0and∂2z/∂x∂x>0,then(x,y)isapointthatzisalocalminimum.If∆<0,then(x,y)isa 88September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book74ReliabilityModelingwithApplicationssaddlepoint.Toobtainoptimalt∗andt∗,wehaveaf∂C2(ta,tf)caha(ta)−αfhf(ta−tf)+cd=,∂tatf+T∂C2(ta,tf)αaha(tf)+αfhf(ta−tf)−cdC2(ta,tf)=−,∂tftf+Ttf+T∂2C(t,t)α[h′(t−t)(t+T)+h(t−t)]−ch(t)−c2afffafffafaaad=,∂ta∂tf(tf+T)2∂2C(t,t)ch′(t)−αh′(t−t)2afaafffaf=,∂t2atf+T∂2C(t,t)αh′(t)−αh′(t−t)−2∂C(t,t)2afaaaffaf2af=.∂t2tf+TfWhenweset∂C2(ta,tf)/∂ta=0and∂C2(ta,tf)/∂tf=0,∂2C(t,t)αh′(t−t)2afffaf=,∂ta∂tf(tf+T)2∂2C(t,t)∂2C(t,t)αh′(t)−αh′(t−t)2af2afaaaffaf==>0,∂t2a∂t2tf+Tf∂2C(t,t)∂2C(t,t)∂2C2af2af2−·∂ta∂tf∂t2a∂t2f2ααh′(t)h′(t−t)−[αh′(t)]2afaafafaaa=<0.(tf+T)2Therefore,whenta>tf,C2(ta,tf)hasalocalminimalpoint.Thus,wecansaythatifthereexists(ta,tf)(ta>tf)satisfies∂C2(ta,tf)/∂ta=0and∂C2(ta,tf)/∂tf=0,ifconditiontf+T>ta>t>0issatisfied,then(t,t)isoptimal(t∗,t∗)whichminimizesthefafafexpectedcostC(t∗,t∗).Further,if(t,t)isnotsatisfiedthecondition,2afafoptimal(t∗,t∗)existsonadditonalconditiont+T=tort=0oraffaatf=0.4NumericalExamplesWecomputenumericallyoptimalreplacementtimes.Especially,wesup-posethatthefreshunithasaWeibulldistributionwithcumulativehazardfunctionH(t)=µts(0 89September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookOptimalReplacementIntervalofaDualSystem75Table1Optimaltainrandomfailureperiodmodelαsf0.70.80.9100.02970.01000.0003150.11480.07580.0187200.29950.31960.3317250.63020.97543.0893301.15722.427019.1283351.93455.245889.3604403.019110.2275100.0000454.470818.4303100.0000506.351931.2119100.00004.1ReplacementinRandomFailurePeriodWecomputenumericallyoptimalt∗whichsatisfiestheequation(4),thatais,t∗minimizestheexpectedcostC(t)in(1).Table1givesoptimalt∗a1aaforαf=10,15,20,25,30,35,40,45,50ands=0.7,0.8,0.9.Wecanseethatweshouldtakealargeintervalofdualsystem,whenadditionalcostαinsinglesystemislarge.Whenαissmall,optimalt∗decreasewhenffashapeparametersincrease.Ontheotherhand,whenαfislarge,optimalt∗increasewhenshapeparametersincrease.Thisreasonisconsideredasafollows.Whenαissmallt∗issmallasmentionedabove.Ift∗islargerfaathan1,h(t∗)=µst∗s−1islargewhensislarge.Ontheotherhand,ift∗faaaissmallerthan1,h(t∗)issmallerwhensislarge.Wecanconsiderthisfatendencygivethenumericalresults.4.2ReplacementinWearoutFailurePeriodWecomputenumericallyoptimalt∗andt∗whichminimizetheexpectedafcostrateC2(ta,ts).Especially,wesupposethattheagingunithasaWeibulldistributionwithcumulativehazardfunctionH(t)=λtm(m>1).aTable2givesoptimalt∗forthesameparameterofTable1whent=10.afBecauset=10,t∗islargerthan10,optimalt∗islargerthanoneinfaaTable1.Wecanseet∗−tinTable2issimilarwithandlargerthanoneafinTable1.Table3giveoptimalt∗forα=10,15,20,25,30,35,40,40,50andfam=1.1,1.2,1.3,1.4,1.5,whenαf=50,s=0.7,ta=50.Wecanseethatt∗whenαislarge,t∗shouldbesmall,i.e.,theintervalofsingleunitaafshouldbesmall. 90September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book76ReliabilityModelingwithApplicationsTable2Optimaltainwearoutfailureperiodwhentf=10αsf0.70.80.91010.030210.010210.00031510.116710.077810.01962010.304610.327710.34872510.640811.000013.24733011.176612.488330.10663511.966915.3782103.93034013.069720.4858110.00004514.545728.8957110.00005016.458442.0000110.0000Table3Optimaltinwearoutfailureperiodwhenαf=50,s=0.7,ta=50fαam1.11.21.31.41.51040.2279338.8273136.2140731.3748424.015301539.0696836.5479431.6295823.4938515.305352037.7292033.7347626.0821216.4754210.028342536.1729630.3226720.2497311.404276.925443034.3620826.3246915.012298.032515.020043532.2531921.9011310.875395.815273.786894526.9680313.211255.723443.310932.359165023.734869.701274.234422.589871.927525ConclusionsWehaveconsideredoptimalreplacementpolicieswithanintervalofdualsystemfortwocases:(1)Agingunitisinrandomfailureperiod,and(2)anagingunitisinwearoutfailureperiod.Wehaveobtainedtheexpectedcostforaintervalthatthefreshunitisininitialfailureperiod.Further,wehavederivedanalyticallyoptimalpoliciesofstoppingagingunitwhichminimizestheexpectedcost.NumericalexampleshavebeengivenwhenthetimestofailureofanagingunitareexponentialorWeibulldistributions.Theseformulationsandresultswouldbeappliedtootherrealsystemssuchasdigitalcircuitsorserversystemsbysuitablemodifications.ReferencesBarlow,R.E.andProschan,F.(1965).MathematicalTheoryofReliability(JohnWiley&Sons,NewYork).LotfiTadj,Mohamed-SalahOuali,SoumayaYacoutandDaoudAit-Kadi(2011).ReplacementModelsWithMinimalRepair(SpringerVerlag,London). 91September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookOptimalReplacementIntervalofaDualSystem77Mizutani,S.andNakagawa,T.(2010).“OptimalMaintenancePolicywithanIntervalofDuplexSystem”,in:S.Chukova,J.HaywoodandT.Dohi(eds),AdvancedReliabilityModelingIV(McGraw-Hill,Taiwan),pp.496–503.Murthy,D.N.P.(1991).“Anoteonminimalrepair”,IEEETransactionsReliability,40,pp.245–246.Murthy,D.N.P.andNguyen,D.G.(1985a).“Studyoftwo-componentsystemwithfailureinteraction”,NavalResearchLogistics32,pp.239–248.Murthy,D.N.P.andNguyen,D.G.(1985b).“Studyofamulti-componentsystemwithfailureinteraction”,EuropeanJournalofOperationsResearch21,pp.330–338.Nakagawa,T.(1981).“ASummaryofperiodicreplacementwithminimalrepairatfailure”,JournalOperationsResearchSocietyJapan24,pp.213–227.Nakagawa,T.(1984).“Optimalnumberofunitsforaparallelsystem”,JournalAppliedProbability21,pp.431–436.Nakagawa,T.(1987).“Optimumreplacementpoliciesforsystemswithtwotypesofunits”,in:S.OsakiandJ.H.Cao(eds),ReliabilityTheoryandAppli-cationsProceedingsoftheChina-JapanReliabilitySymposium(Shanghai,China).Nakagawa,T.(2005).MaintenanceTheoryofReliability(SpringerVerlag,London).Nakagawa,T.(2011).StochasticProcesseswithApplicationstoReliabilityTheory(SpringerVerlag,London).Osaki,S.(1992).AppliedStochasticSystemModeling(SpringerVerlag,Berlin).Pham,H.,Suprasad,A.andMisra,B.(1996).“ReliabilityandMTTFpredictionofk-out-of-ncomplexsystemswithcomponentssubjectedtomultiplestagesofdegradation”,InternationalJournalSystemScience27,pp.995–1000.Yasui,K.,Nakagawa,T.andOsaki,S.(1988).“Asummaryofoptimumreplace-mentpoliciesforaparallelredundantsystem”,MicroelectronReliability28,pp.635–641. 92May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank 93September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookChapter5CumulativeDamageModelswithRandomWorkingTimesXufengZhao1,2,CunhuaQian1andShey-HueiSheu31SchoolofEconomicsandManagement,NanjingUniversityofTechnology,30PuzhuRoad,Nanjing211816,China2GraduateSchoolofManagementandInformationSciences,AichiInstituteofTechnology,1247Yachigusa,Yakusa-cho,Toyota470-0392,Japan3DepartmentofStatisticsandInformaticsScience,ProvidenceUniversity,200ChungChiRd.,Shalu,Taichung43301,Taiwan1IntroductionMostoperatingunitsarerepairedorreplacedwhentheyhavefailed.How-ever,itmayrequiremuchtimeandsufferhighercosttorepairafailedunitorreplaceitwithanewone,sothatitisnecessarytoreplaceittopreventfailures.Replacementafterfailureandbeforefailurearecalledcorrectivereplacement(CR)andpreventivereplacement(PR),respectively.There-centpublishedbooks[Osaki(2002);Pham(2003);Nakagawa(2005);WangandPham(2007);KobbacyandMurthy(2008);Manzini,Regattieri,PhamandFerrari(2010);Nakagawa(2011)]collectedmanyPRmodelsorothermaintenancemodelsintheoryandtheirapplicationsinindustrialsystems.Especially,Nakagawa(2007)summarizedsufficientlymaintenancepoliciesandtheiroptimizationproblemsforshockanddamagemodelswhicharecalledcumulativedamagemodels.Thesemodels,whichplayanimportantroleinreliabilitytheory,areconsideredasasequenceofshocksthatoccurrandomlyintimeandgivesomeamountofdamagetoaunit.Thedamageisaccumulatedtothecurrentdamagelevel,weakenstheunitgradually,andmakesitfailurewhenthetotaldamageexceedsafailurelevel.79 94September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book80ReliabilityModelingwithApplicationsSomeunitsinofficesandindustriesexecutejobsorcomputerprocedureswithrandomworkingtimessuccessively.Forsuchunits,itwouldbeimpos-sibleorimpracticaltomaintainorreplacetheminastrictregularfashion,e.g.,aplannedtimeT,eventhoughthemaintenanceorreplacementtimecomes,becausesuddensuspensionofthejobmaysufferlossesofproductionindifferentdegreesifthereisnosufficientpreparationinadvance[BarlowandProschan(1965),p.72;Nakagawa(2005),p.245].Ontheotherhand,wesometimesareinterestedincertainquantitiesofunitsinwhichtoan-alyzetheirreliabilitiesandmaintenancesthatareusuallyobservedbyauniquetimescalesuchasageoroperatingtime,buttheyareoftenmea-suredbysomecombinedscalesinreliabilityapplications.Alternativetimescaleswereinvestigatedandhowtoselecteffectiveonestoanalyzemeantimetofailure(MTTF)wasdiscussed[DuchesneandLawless(2000)].Re-placementpolicieswithtwotimescales,suchasageandnumberofuses,wereproposed[Nakagawa(2005),p.83].Takingpartsofanaircraftasanexample,appropriatemaintenanceandreplacementpoliciesareusuallyscheduledatatotalhoursofoperationsorataspecifiednumberofflightssincethelastmajoroverhaul[Nakagawa(2008),p.149].Byconsideringthefactorsofworkingtimesinoperations,thereliabilityquantitiesoftherandomagereplacementpolicywereobtained[BarlowandProschan(1965),p.72].Severalschedulesofjobsthathaverandompro-cessingtimesweresummarized[Pinedo(2002)].Whenajobhasavariableworkingcycleorprocessingtime,itwouldbebettertodomaintenanceorreplacementafterthejobisjustcompletedeventhoughthemainte-nancetimehasarrived[Sugiura,MizutaniandNakagawa(2004);Naka-gawa(2005),p.245].Fromsuchaviewpoint,ageandperiodicreplacementpolicieswhicharedoneatthefirstcompletionofsomeworkingtimeoveraplannedtimeTwasproposed[Chen,MizutaniandNakagawa(2010);Chen,NakamuraandNakagawa(2010)],andoptimalagereplacementwithovertimepolicywasderivedin[Zhao,QianandNakagawa(2012)].Furthermore,bycombiningaplannedreplacementwithworkingtimes,ageandperiodicreplacementpolicies,wheretheunitisreplacedataplannedtimeTandattheNthrandomworkingtime,werediscussed[Chen,MizutaniandNakagawa(2010);Chen,NakamuraandNakagawa(2010)].Byconsideringthecaseswhenreplacementcostssufferedforfailureswouldbeestimatedtobenotsohighandthefactorofworkingtimes,ageandperiodicreplacementpolicies,wheretheunitisreplacedataplannedop-eratingtimeToratarandomworkingcycleY,whicheveroccurslast,wereproposed[ZhaoandNakagawa(2012)].Thisalsoindicatedthatpoli- 95September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookCumulativeDamageModelswithRandomWorkingTimes81ciesproposedin[Chen,MizutaniandNakagawa(2010);Chen,NakamuraandNakagawa(2010)]wouldcausefrequentandunnecessaryreplacementwhichmayincurproductionlossesundertheassumptionof“whicheveroc-cursfirst”.Thenotion“whicheveroccurslast”wasappliedinacumulativedamagemodel,wheretheunitisreplacedbeforefailureataplannedtimeT,atashocknumberN,oratadamagelevelZ,wasdiscussed[Zhao,NakayamaandNakamura(2011);Zhao,QianandNakagawa(2013)].Fromaboveconsiderations,thischapterconsidersanoperatingunitwhichworksatsuccessiverandomtimesforjobsanditsreplacementpoli-cies,usingthecumulativedamagemodels[Nakagawa(2007)]byreplacingshockwithwork:Eachwork,whichhasavariableworkingtime,causessomedamagetotheunitandthesedamageareadditive,andthesystemfailswhenthetotaldamagehasexceededafailurelevelKandcorrectivereplacementismade,orfailswithprobabilityp(x)whenthetotaldamageisxandminimalrepairismade.Asthepreventivereplacementpolicy,(1)theunitisreplacedbeforefailureatNthworkingtimeforthefirstmodel;(2)theunitisreplacedatthefirstcompletionofworkingtimeafterThasarrived;(3)theunitisreplacedatthefirstcompletionofworkingtimewhenthetotaldamagehasexceededathresholdlevelZ;(4)theunitisreplacedbeforefailureattimeTofoperationsoratnumberNofwork-ingtimes,whicheveroccursfirstandlast.TwocaseswherefailurelevelKandfailureprobabilityp(x)havebeenconsideredin(2)and(3).Expectedcostratesofeachmodelareobtainedandoptimalreplacementpoliciesarediscussedanalyticallyandcomputednumerically.2ModelingandOptimizationItisassumedthatXj(j=1,2,···)aresuccessiveworkingtimesoftheunitandareindependentandhaveanidenticaldistributionF(t)≡Pr{Xj≤t}withafinitemean1/λ.Thatis,thesystemworksatarenewalprocesswithitsdistributionF(t).Supposethateachworkofthejobincurssomedamagetotheoperatingunitandthetotaldamageisadditive,whichiscalledacumulativedamagemodel[Nakagawa(2007),p.16].Thatis,thejthworkcausessomedamagetotheunitintheamountYj(j=1,2,···){}accordingtoanidenticaldistributionG(x)≡PrYj≤xwithafinitemean1/µ.Then,theprobabilitythatjtimesofworksarecompletedin(0,t]is,from[Nakagawa(2007),p.17],Pr{N(t)=j}=F(j)(t)−F(j+1)(t)(j=0,1,2,···), 96September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book82ReliabilityModelingwithApplicationsandthedistributionofthetotaldamageZ(t)attimetis∑∞Pr{Z(t)≤x}=G(j)(x)[F(j)(t)−F(j+1)(t)],j=0whereΦ(j)(t)denotesthej-foldStieltjesconvolutionofΦ(t)withitselfand(0)∑∞(j)Φ(t)≡1fort>0foranyfunctionΦ(t),andM(x)≡G(x)j=1whichistherenewalfunctionofG(x).2.1WorkingNumberNItisassumedthattheoperatingunitfailswhenthetotaldamagehasex-ceededafailurelevelK,anditsfailureisdetectedandreplacementismadeatthecompletionofworkingtime.Asthepreventivereplacementpolicy,theunitisreplacedbeforefailureatNth(N=1,2,···)workingtime.Then,theexpectedcostrateis[Nakagawa(2007),p.44]C(N)c−(c−c)G(N)(K)KKN=∑(N=1,2,···),(1)λN−1G(j)(K)j=0wherecKandcNaretherespectivereplacementcostsatfailureandtheNthworkingtimewithcK>cN.FromtheinequalityC(N+1)−C(N)≥0,anoptimalnumberN∗whichminimizestheexpectedcostrateC(N)in(1)satisfiesN∑−1(j)(N)cNL(N)G(K)−[1−G(K)]≥(N=1,2,···),(2)cK−cNj=0whereG(N)(K)−G(N+1)(K)L(N)≡.G(N)(K)IfL(N)increasesstrictlywithN,thentheleft-handsideof(2)alsoin-creasesstrictlywithN.Therefore,ifL(∞)[1+M(K)]>cK/(cK−cN),thenthereexistsauniqueandminimumN∗(1≤N∗<∞)whichsatisfies(2).Inparticular,whenG(x)=1−e−µx,i.e.,G(j)(x)=∑∞i−µx[(µx)/i!]e,fromExample2.2[Nakagawa(2007),p.24],L(N)=i=j∑[(µK)N/N!]/∞[(µK)i/i!]increasesfrome−µKto1.Then,theresult-i=NingcostrateisC(N∗)(c−c)L(N∗)<≤(c−c)L(N∗+1).(3)KNKNλ 97September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookCumulativeDamageModelswithRandomWorkingTimes83Table1presentsoptimalN∗whichsatisfy(2)andC(N∗)/λfordifferentµKandc/(c−c).Clearly,N∗increasewithbothµKandc/(c−NKNNKcN).Thatis,tocontroleffectivelyahighcostsufferedforfailure,wemustmakethereplacementtimeearlierasafailurelevelKislowerorafailurereplacementcostcKishigher.Table1OptimalNandC(N)/λforµKandcN/(cK−cN)cNµK=15µK=20µK=30cK−cNNC(N)/λNC(N)/λNC(N)/λ0.0160.002190.0013150.00070.0370.0054100.0035170.00200.0570.0082100.0055170.00320.0780.0110110.0074180.00430.1080.0148110.0101190.00590.30100.0372130.0262210.01600.50110.0569140.0407220.02530.70120.0753150.0542230.03411.00120.1009160.0734240.04692.2ReplacementOverTimeTorLevelZ2.2.1OvertimeT1.PolicyIAunitisreplacedbeforeaplannedtimeT(0≤T≤∞)whenthetotaldamagehasexceededafailurelevelK,andafterT,itisreplacedatthefirstcompletionofworkingtime.Then,theexpectedcostrateis[Nakagawa(2007),p.54]c−(c−c)∑∞[F(j)(T)−F(j+1)(T)]G(j+1)(K)C1(T)KKTj=0=∑∞,(4)λF(j)(T)G(j)(K)j=0wherecTisreplacementcostatthecompletionofworkingtimeafterT.WefindanoptimaltimeT∗whichminimizestheexpectedcostrate1C(T)in(4)whenF(t)=1−e−λtandG(x)=1−e−µx,i.e.,F(j)(t)=∑1∑∞[(λt)i/i!]e−λtandG(j)(x)=∞[(µx)i/i!]e−µx.Differentiatingi=ji=jC1(T)withrespecttoTandsettingitequaltozero,∑∞∑∞(λT)jcQ(T)F(j)(T)G(j)(K)−e−λT[1−G(j+1)(K)]=T,j!cK−cTj=0j=0(5) 98September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book84ReliabilityModelingwithApplicationswhere∑∞jj+1−µK[(λT)/j!][(µK)/(j+1)!]ej=0Q(T)≡∑∞.[(λT)j/j!]G(j+1)(K)j=0()BecauseQ(T)increasesstrictlywithTfromµK/eµK−1to1,theleft-handsideof(5)alsoincreasesstrictlyfromµK−1+e−µKµKQ≡≤eµK−12toµK.Therefore,wehavethefollowingoptimalpolicies:()1.IfQ≥c/c−c,thenT∗=0,i.e.,theunitisreplacedatthefirstTKT1completionofworkingtime,andtheexpectedcostrateisC1(0)()=cK−cK−cTG(K).(6)λ()2.IfQ 99September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookCumulativeDamageModelswithRandomWorkingTimes85Table2OptimalλTandC1(T)/λforµKandcT/(cK−cT)11cTµK=15µK=20µK=30cK−cTλTC1(T)/λλTC1(T)/λλTC1(T)/λ1111110.014.040.00326.370.001911.740.00100.035.160.00757.750.004713.570.00250.055.790.01118.520.007214.560.00400.076.260.01449.080.009415.270.00530.106.810.01909.730.012616.090.00720.308.960.044012.220.030619.150.01850.5010.290.064713.740.046020.960.02840.7011.350.083314.920.060122.350.03771.0012.670.108716.370.079524.030.0508completionofworkingtime.Then,theexpectedcostrateis[Nakagawa(2007),p.98]∑∞(j)∗jcMj=1F(T){1−G(θ)}+cTC2(T)=∫∞∑∞∫T∫∞(j),(9)T+F(t)dt+[F(u)du]dF(t)Tj=10T−twherecMiscostsufferedforminimalrepairateachfailure,and∫∞∫∞p(x)dG(j)(x)=(1−e−θx)dG(j)(x)=1−[G∗(θ)]j,00whichmeanstheprobabilitythattheunitfailsattheendofjthwork.Here,G∗(θ)denotestheLaplace-StietjestransformofG(x),i.e.,G∗(θ)≡∫∞e−θxdG(x)<1forθ>0.0DifferentiatingC2(T)withrespecttoTandsettingitequaltozero,∗∗−λT[1−G∗(θ)]G(θ)−λT[1−G∗(θ)]cT1−(1+λT)G(θ)e+{1−e}=.1−G∗(θ)cM(10)Theleft-handsideof(10)increasesstrictlywithTfrom1−G∗(θ)to1/[1−G∗(θ)].Therefore,wehavethefollowingoptimalpolicies:1.If1−G∗(θ)≥c/c,thenT∗=0.TM22.If1−G∗(θ) 100September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book86ReliabilityModelingwithApplicationsSupposethatG(x)=1−e−µx,Table3presentsoptimalλT∗which2satisfies(10)andC(T∗)fordifferentθandc/cwhenµ=1.0.Itis22TMshownthatλT∗decreaseswithθbecausetheoperatingunithasahigher2failureprobabilitywhenθbecomeslarger,andλT∗increaseswithc/c2TMbecauseminimalrepaircostsufferslessthanbeforesothattheunitcouldbeoperatingforalongertime.Table3OptimalλTandC2(T)forθandcT/cMwhenµ=1.0.22θ=0.01θ=0.03θ=0.05cT/cMλTC2(T)λTC2(T)λTC2(T)2222220.13.460.04301.480.06990.840.08460.36.950.07523.630.12632.600.15850.59.380.09705.130.16353.830.20600.711.380.11466.370.19324.850.24351.013.940.13667.980.22996.170.28963.025.940.232615.820.386812.860.48315.034.790.296522.010.487718.460.60417.042.390.347127.660.565323.930.694710.052.540.409135.850.657332.640.79822.2.2OverlevelZ1.PolicyIAunitisreplacedwhenthetotaldamagehasexceededafailurelevelK,andisalsoreplacedpreventivelyatthenextworkingtimewhenthedamageisbetweenZandK.Then,theexpectedcostrateis[Nakagawa(2007),p.56]∫KcK−(cK−cZ){ZG(K−x)dG(x)∫Z∫K−xC1(Z)+0[Z−xG(K−x−y)dG(y)]dM(x)}=∫,(12)λ1+G(K)+ZG(K−x)dM(x)0wherecZisreplacementcostatthecompletionofworkingtimewhenthedamageisbetweenZandK.WhenG(x)=1−e−µx,differentiatingC(Z)withrespecttoZand1settingitequaltozero,[]−µ(K−Z)µ(1+µZ)(K−Z)cZe−1=.(13)1−e−µ(K−Z)cK−cZTheleft-handsideof(13)increasesstrictlywithZfromQtoµK.There-fore,wehavethefollowingoptimalpolicies: 101September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookCumulativeDamageModelswithRandomWorkingTimes871.IfQ≥c/(c−c),thenZ∗=0,andtheexpectedcostrateisZKZ1C(0)c−(c−c)[1−(1+µK)e−µK]1KKZ=.(14)λ2−e−µK2.IfQ 102September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book88ReliabilityModelingwithApplicationsDifferentiatingC2(Z)withrespecttoZandsettingitequaltozero,∫∞∫Z[∫∞][2+M(Z)]p(Z+x)dG(x)−p(x+y)dG(y)dM(x)000∫∞cZ−p(x)dG(x)=.(17)0cMDenoteD(Z)betheleft-handsideof(17),thenD(Z)increasesstrictly∫∞withZfromp(x)dG(x)toD(∞).Therefore,wehavethefollowing0optimalpolicies:∫∞1.Ifp(x)dG(x)≥c/c,thenZ∗=0,and0ZM2∫∞C2(0)cM0p(x)dG(x)+cZ=.(18)λ2∫∞2.If0p(x)dG(x) 103September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookCumulativeDamageModelswithRandomWorkingTimes892.3ReplacementFirstandLast2.3.1ReplacementfirstItisassumedthatanoperatingunitfailswhenthetotaldamageexceedsathresholdlevelK(0 104September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book90ReliabilityModelingwithApplicationsTherefore,ifVF(∞,N)>cP/(cK−cP),thenthereexistsafiniteanduniqueT∗(0 105September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookCumulativeDamageModelswithRandomWorkingTimes91occurslast.Then,theexpectedcostrateis[Zhao,QianandNakagawa(2013)]c+(c−c){1−∑∞F(j+1)(T)[G(j)(K)−G(j+1)(K)]}CL(T,N)PKPj=Nλ=∑N−1(j+1)∑∞.F(T)G(j)(K)+F(j+1)(T)G(j)(K)j=0j=0(28)DifferentiatingCL(T,N)withrespecttoTforNandsettingitequaltozero,N∑−1∑∞(j+1)(j)(j+1)(j)RL(T,N)F(T)G(K)+F(T)G(K)j=0j=0∑∞(j+1)(j)(j+1)cP+F(T)[G(K)−G(K)]−1=,(29)cK−cPj=Nwhere∑∞f(j+1)(t)[G(j)(K)−G(j+1)(K)]j=NRL(t,N)≡∑∞(j+1)(j),f(t)G(K)j=Nandf(j+1)(t)≡[λ(λt)j/j!]e−λt.WhenG(j)(K)=∑∞[(µK)i/i!]e−µK,iti=jisprovedfrom[Zhao,QianandNakagawa(2013)],thatRL(t,N)increasesstrictlywithtfrom1−G(N+1)(K)/G(N)(K)to1.Denotingtheleft-handsideof(29)byVL(T,N)andrL(t,N)≡dRL(t,N)/dt,thendVL(T,N)/dTisN∑−1∑∞(j+1)(j)(j+1)(j)>0,(30)rL(T,N)F(T)G(K)+F(T)G(K)j=0j=0whichfollowsthatVL(T,N)increasesstrictlywithTfromN∑−1(j)(N)VL(0,N)=L(N)G(K)−G(K)j=0toVL(∞,N)=M(K).Therefore,ifVL(0,N) 106September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book92ReliabilityModelingwithApplicationsFormingtheinequalityCL(N+1)−CL(N)≥0,∑∞∑∞c(j+1)(j)(j)PF(T)G(K)[L(j)−L(N)]+L(N)G(K)−1≥,cK−cPj=Nj=0(32)whereL(j)(j=0,1,2,···)isgivenin(2).From[Nakagawa(2007),p.24],G(j+1)(K)/G(j)(K)decreasesstrictlywithjwhenG(j)(K)=∑∞[(µK)i/i!]e−µK,andL(j)increasesstrictlyi=jwithjfrome−µKto1.Denotingtheleft-handsideof(32)byV(N,T),LVL(N+1,T)−VL(N,T)is∑∞N∑−1[R(N+1)−R(N)]F(j+1)(T)G(j)(K)+G(j)(K)>0,(33)j=Nj=0whichfollowsthatVL(N,T)increasesstrictlywithNfrom∑∞∫KF(j+1)(T)[G(K)−G(K−x)]dG(j)(x)<0j=00toM(K).Therefore,ifM(K)>cP/(cK−cP),thenthereexistsauniqueandminimumN∗(1≤N∗<∞)whichsatisfies(32),andtheresultingcostLLrateisC(T,N∗)(c−c)L(N∗−1) 107September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookCumulativeDamageModelswithRandomWorkingTimes93replacementfirstshouldbeadoptedwhenN≥N∗andV(N)>cP/(cK−cP),andthepolicyinSectionWorkingNumberNshouldbeadoptedwhenN≥N∗andV(N)≤c/(c−c).PKPTable6OptimalT,T,andtheircostrateswhenλ=1,µ=1andK=10LFcPN=5N=8cK−cPTCL(T,N)TCF(T,N)TCL(T,N)TCF(T,N)NLLFFLLFF0.10.000.026∞0.0260.000.0414.750.03350.21.280.046∞0.0460.000.0546.840.05260.33.560.065∞0.0660.000.0679.490.06660.45.010.084∞0.0860.000.08014.090.07970.56.090.100∞0.1060.000.09328.350.09270.66.970.115∞0.1260.000.106∞0.10680.77.750.129∞0.1460.000.119∞0.11980.88.440.142∞0.1660.000.131∞0.13180.99.070.155∞0.1860.000.144∞0.14481.09.680.167∞0.2062.620.157∞0.1579Table6indicates:•ThereexistthreecasesbetweenT∗andT∗accordingtoN∗:0 108September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book94ReliabilityModelingwithApplicationswhenN=5,preventivereplacementisdoneat5workingtimesforreplacementfirst,nomatterwhethercK/cPislargeorsmall;how-ever,preventivereplacementwouldbedonearoundfromλT∗≈0toLλT∗≈10workingtimesaccordingtoc/c.ItseemmorereasonableLKPwhenreplacementlastisadoptedforsuchcasestoavoidunnecessaryreplacements.Next,wecompareN∗in(26)withN∗in(34)todecidewhichpolicyisFLbetterforagivenT.WhenN(K)>cP/(cK−cP),comparetheleft-handsideof(24)and(32)bydenotingA(N)≡VL(N,T)−VF(N,T).ItcanbeeasilyshownthatA(N+1)−A(N)isN∑−1∑∞(j+1)(j)(j+1)(j)>0,[L(N+1)−L(N)]F(T)G(K)+F(T)G(K)j=0j=NwhichfollowsthatA(N)increaseswithNstrictlyto∑∞(j)(j)A(∞)=F(T)G(K)>0.j=0Thus,thereexistsauniqueandminimumN∗(1≤N∗<∞)whichsatisfiesAAA(N)≥0.From(24),denotingthat∗N−1∑AH(N∗)≡F(j+1)(T)G(j)(K)[L(N∗)−L(j)].(35)AAj=0Then,thefollowingcomparisonresultscanbegiven:1:IfH(N∗) 109September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookCumulativeDamageModelswithRandomWorkingTimes95Table7OptimalN,N,andtheircostrateswhenλ=1,µ=1andK=10LFcPT=5T=8cK−cPNCL(T,N)NCF(T,N)NCL(T,N)NCF(T,N)NLLFFLLFF0.150.03250.02960.04950.02650.260.04860.05260.06360.04960.360.06470.07470.07560.06460.470.07870.09570.08870.08070.570.09380.11670.10070.09670.680.10680.13780.11280.11180.780.11990.15780.12480.12680.880.13290.17880.13680.14180.990.145100.19890.14890.15681.090.156100.21990.15990.1699N68AH(N)0.2700.821AH(N−1)0.1420.547Alastshouldbeadopted,e.g.,whenT=5,H(N∗)=0.270whichisAlessthanc/(c−c)=0.3,thenC(T,N∗) 110September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book96ReliabilityModelingwithApplications3ConclusionsWehavediscussedfourreplacementpoliciesforanoperatingunitwhichworksatsuccessiverandomtimesforjobs,wheretheunitfailsduetodam-agethatcanbeadditivecausedbyjobs.Usingthetechniqueofcumulativedamagemodels,expectedcostrateshavebeenobtained,andtheoptimalreplacementpolicieshavebeendiscussedanalytically.Numericalexampleshavebeencomputedforallmodelsandsomeusefulexplanationshavebeengiven.ThepoliciesinSections2.1and2.2aremadeonlyattheendofsomeworkcompletion,althoughtheperformancesforthesethreepoliciesaredifferent.Itseemsmorepracticaltoperformreplacementwhentheunithascompletedthejobwithoutstops.InSection2.1,wecouldconsiderthereplacementatworkingnumberNasastandardpolicy,whichisasinglepolicythathasbeenonlyconsideredthefactorofjob.Whenweneedtotakethefactors,e.g.,totaloperationtimeanddamagelevel,intoconsider-ation,themodelsproposedinSections2.2.1and2.2.2aremorepracticaltoperformandtwocaseswherefailurelevelKandfailureprobabilityp(x)havebeenconsidered.Fromnumericalcomparisonsinthesetwosections,theoverlevelZpoliciesarebetterthanthoseinovertimeTfromtheeconomicalpoint,becauseoverlevelZpoliciescouldmonitorthedamagelevelmoreaccuratelythatwouldcausethefailureoftheunit.ThepoliciesinSection2.3arecombinedaplannedreplacementwithworkingtimes,i.e.,timeTofoperationsandnumberNofworkingtimes.Twoperformancemechanisms“whicheveroccursfirst”and“whicheveroc-curslast”aremodeled,wecalledthemreplacementfirstandlast,respec-tively.Asanewlyproposednotion“whicheveroccurslast”,themostin-terestingpointisthatwehavealreadyfoundthecaseswhenreplacementlastshouldbeadoptedornot.Finally,wegiveapotentialapplicationofsuchmodelstomaintainadatabaseincomputerscience:(i)Normally,thedatabaseismaintainedatperiodictimessuchasday,week,month,etc.However,itisnecessarytoguaranteeACID(atomicity,consistency,isolation,durability)propertiesofdatabasetransactions[HaerderandReuter(1983);GrayandReuter(1992);Lewis,BernsteinandKifer(2002)],sothatitisnotadvisabletosuspendanytransactionwhenitisunderbusystate.(ii)Cumulativedamagemodelshavebeensuccessfullyformulatedtheincrementalprocessesofupdateddatainadatabase,[Qian,PanandNakagawa(2002)].Inotherwords,wecanmonitorthecumulativeupdateddataatanytime. 111September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookCumulativeDamageModelswithRandomWorkingTimes97ReferencesBarlow,R.E.andProschan,F.(1965).MathematicalTheoryofReliability(Wiley,NewYork).Chen,M.,Mizutani,S.andNakagawa,T.(2010).Randomandagereplacementpolicies,InternationalJournalofReliability,QualityandSafetyEngineer-ing17,pp.27–39.Chen,M.,Nakamura,S.andNakagawa,T.(2010).Replacementandpreventivemaintenancemodelswithrandomworkingtimes,IEICETrans.Fundamen-talsE93-A,pp.500–507.Duchesne,T.andLawless,J.(2000).Alternativetimescalesandfailuretimemodels,LifetimeDataAnalysis6,pp.157–179.Gray,J.andGray,A.(1992).TransactionProcessing:ConceptsandTechniques(MorganKaufmann,USA)Haerder,T.andReuter,A.(1983).Principlesoftransaction-orienteddatabaserecovery,ACMComputingSurveys15,pp.287–317.Kobbacy,K.A.H.andMurthy,D.N.P.(2008).ComplexSystemMaintenanceHandbook(Springer,London).Lewis,M.L.,Bernstein,B.andKifer,M.(2002).DatabasesandTrans-actionProcessing:AnApplication-OrientedApproach(AddisonWesley,USA).Manzini,R.,Regattieri,A.,Pham,H.andFerrari,E.(2010).MaintenanceforIndustrialSystems(Springer,London).Nakagawa,T.(2005).MaintenanceTheoryofReliability(Springer,London).Nakagawa,T.(2007).ShockandDamageModelsinReliabilityTheory(Springer,London).Nakagawa,T.(2008).AdvancedReliabilityModelsandMaintenancePolicies(Springer,London).Nakagawa,T.(2011).StochasticProcesswithApplicationstoReliabilityTheory(Springer,London).Osaki,S.(2002).StochasticModelsinReliabilityandMaintenance(Springer,Berlin).Pinedo,M.(2002).SchedulingTheory,AlgorithmsandSystems(PrenticeHall,EnglewoodCliffs,NJ).Pham,H.(2003).HandbookofReliabilityEngineering(Springer,London).Qian,C.,Pan,Y.andNakagawa,T.(2002).Optimalpoliciesforadatabasesystemwithtwobackupschemes.RAIRO-OperationsResearch36,pp.227–235.Sugiura,T.,Mizutani,S.andNakagawa,T.(2004).Optimalrandomreplacementpolicies,inProceedingsoftheTenthISSATInternationalConferenceonReliabilityandQualityinDesign(LasVegas,Nevada),pp.99–103.Wang,H.andPham,H.(2007).ReliabilityandOptimalMaintenance(Springer,London).Zhao,X.,Nakayama,K.andNakamura,S.(2011).Cumulativedamagemod-elswithreplacementlast,inInternationalConferencesASEA/DRBC/EL, 112September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book98ReliabilityModelingwithApplicationsCommunicationsinComputerandInformationScience257(JejuIsland,Korea),pp.338–345.Zhao,X.,Qian,C.andNakagawa,T.(2012).Agereplacementwithovertimepolicy,inThe5thAsia-PacificInternationalSymposiumonAdvancedRe-liabilityandMaintenanceModelingV(Nanjing,China),pp.661–668.Zhao,X.andNakagawa,T.(2012).Optimizationproblemsofreplacementfirstorlastinreliabilitytheory,EuropeanJournalofOperationalResearch223,pp.141–149.Zhao,X.,QianC.andNakagawa,T.(2013).Optimalpoliciesforcumulativedamagemodelswithmaintenancelastandfirst,ReliabilityEngineering&SystemSafety110,pp.50–59. 113September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookPART2ReliabilityAnalysis99 114May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank 115September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookChapter6ModulesofMulti-StateCoherentSystems—OrderTheoreticalRelationsFumioOhi1OmohiCollege,NagoyaInstituteofTechnology,Gokiso-cho,Showa-ku,Nagoya,466-8555,Japanemail:ohi.fumio@nitech.ac.jp1IntroductionAbasicproblemofreliabilitytheoryistoexplainrelationshipsamongthereliabilityperformancesofsystemsandthecomponents.UsingBooleanfunctions,[Mine(1959)]introducedtheconceptofmonotonesystems,whereallthestatespaceswereassumedtobe{0,1},socalledbinarystatesystems,where0and1denotethefailureandthefunctioningstates,re-spectively.Mathematicalaspectsofbinarystatesystemshavebeenfullyexaminedby[BirnbaumandEsary(1965)],[Birnbaumetal.(1961)]and[EsaryandProschan(1963)].[BarlowandProschan(1975)]havesumma-rizedthereliabilitystudiesofthebinarystatemonotonesystems.Inmanypracticalsituations,however,systemsandtheircomponentscouldtakemanyotherperformancelevelsfromtheperfectlyfunctioningstatetothecompletefailurestateandsometimesforsometwostateswecannotsaywhichstateisbetter/worsethananotherstate,thereforeweneedmulti-statereliabilitymodelswithpartiallyorderedstatespacestounderstandandsolvepracticalproblemsasreliabilityestimationandriskanalysis.1CorrespondingAuthor,Tel.:+81-52-735-5393101 116September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book102ReliabilityModelingwithApplicationsMulti-statesystemswereintroducedinthecontextofcannibalizationby[Hirsch,MeisnerandBoll(1968)]and[Hochberg(1973)].Afterthiswork,mathematicalstudiesofmulti-statesystemswerecarriedoutbymanyau-thors;firstmakinguseoftheminimalpathandcutsetsofbinarystatesys-tems,[BarlowandAlexander(1978)]definedmulti-statecoherentsystems.[El-Neweihi,ProschanandSethuraman(1978)]definedthemulti-statesys-temsassumingallthestatespacestobecommonly{0,1,···,M}.[Huang,ZuoandFang(2003)]introducedthemulti-stateconsecutivek-out-of-nsystemsandprovidedalgorithmstoevaluatetheperformanceprobabili-tiesofthesystems,assumingthestatespacestobethesamefinitetotallyorderedsetsas[El-Neweihi,ProschanandSethuraman(1978)].[OhiandNishida(1983)],[Ohi(2010)]havedefinedmulti-statesys-tems,ofwhichstatespacesarefinitetotallyorderedsetshavingnotnec-essarilythesamecardinalnumber,wheretheyhavefullyexaminedorderandstochasticpropertiesofthesystems.[Yu,KorenandGuo(1994)]pre-sentedamodelofmulti-statesystemsunderanassumptionforthestatespacestobepartiallyorderedsets.Amathematicalgeneralizationtopartiallyorderedsetcases,aimingatgivingatheoreticalframeworkofmulti-statesystemsisgivenby[Ohi(2011)]showinganexistencetheoremofseriesandparallelsystemsandadecompositionofsystemsbyseriesandparallelsystems,whichiscalledinthebinarycaseas“max−min”formulaewithminimalpathorcutsets,see[BarlowandProschan(1975)].Aconceptofmodulesorsub-systemsplaysacrucialroleforevaluatingreliabilityperformancesofsystems,sincepracticalsystemsareusuallycom-posedofmodulesandthecomponentsarearrangedtoformthesemodules.Aconceptofmoduleofmulti-statesystemsforthecaseofpartiallyorderedsetshasbeengivenby[Ohi(2012)],acontinuationofourrecentworkof[Ohi(2011)],wheresomebasicordertheoreticalpropertiesarepresentedwithoutproofs.Inthisarticle,weshowperfectproofsofthepropositionsof[Ohi(2012)].Inthispaper,wepresentadefinitionofmodulesofmulti-statesystemsforthecaseofpartiallyorderedstatespacesandshowsomeordertheoreticalpropertiesofthem,whicharethoughttobebasicforfurtherexaminationsofmodulesasgivingstochasticboundsforthesystemsbymodules.Bytheseproperties,itissuggestedthatnormalpropertyofsystemsplayanimportantroleforthedefinitionofmodules.Sinceifthenormalpropertyisnotassumedinthedefinitionofsystems,anysubsetofthecomponentscouldbeamoduleinthecontextofpartiallyorderedsetsandnobrakeis 117September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookModulesofMulti-StateCoherentSystems—OrderTheoreticalRelations103appliedtothedefiningmodules,inotherwords,thedefinitionofmoduleswithoutnormalpropertyismeaningless.Itiswellknowninthecaseofbinarystateseriesandparallelsystemsthatanysubsetofcomponentsconsistingthesystemcanbeamodule.Weshowthesimilarresultinthecaseoftotallyorderedstatespaces,followingthedefinitionofseriesandparallelsystemsof[Ohi(2010)].Wealsopresentonlydefinitionsofak-out-of-nandaconsecutivek-out-of-nmulti-statesystems,ofwhichpreciseexaminationsareremainedforfuturework.NotationsInthispaperweusethefollowingnotations.AfinitesetC={1,2,···,n},Ωi(i∈C)andSareinterpretedrespectivelyasthesetofthecomponents,thestatespacesoftheithcomponentandthesystem.ThedefinitionofasystemispresentedinDefinition6.1.∏1)ForA⊂C,ΩA=i∈AΩiistheproductsetofΩi(i∈A).2)AnelementofΩ(A⊂C)isdenotedbyxAandalsosimplyxA=xAifthereisnoconfusion.WhenA=C,x∈ΩCispreciselywrittenasx=(x1,···,xn),xi∈Ωi(i=1,···n).3)Let{Bj|1≤j≤m}beapartitionofA⊂C.Thenforxj∈∏i∈BjΩi(1≤j≤m),x=(x1,···,xm)isanelementofΩAsuchthatPx=x.Thenforeveryx∈Ω(A⊂C),x=(xB1,···,xBm),ΩBjjAwherexBi=P(x)(i=1,···,m).PistheprojectionfromΩtoBiBiAΩBi.4)(ki,x)∈ΩA(A⊂C,i∈A)isanelementofΩAsuchthatk∈Ωiand∏x∈j∈A\{i}Ωj.5)Forx∈ΩC,wesometimesuse(ki,x)=(x1,···,xi−1,k,xi+1,···,xn)tostressthatthestateoftheithelementofxisk.2Multi-stateCoherentSystemsDefinition6.1.(DefinitionofaMulti-stateSystem)Asystemcomposedofncomponentsisatriplet(ΩC,S,φ)satisfyingthefollowingconditions.(1)C={1,···,n}isthesetofintegersfrom1ton,whereeachnumberistheindexofeachunit.(2)Ωi(i∈C)isafinitelatticesethavingtheleastandthegreatestelementsdenotedbymiandMi,respectively. 118September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book104ReliabilityModelingwithApplications∏n(3)ΩC=i=1ΩiistheproductlatticesetofΩi(i∈C).Anelementx=(x1,···,xn)∈ΩC,calledastatevector,meansacombinationofstatesofthecomponentsasxi∈Ωiisthestateoftheithcomponent.(4)SisafinitelatticesethavingtheleastandthegreatestelementsdenotedbymandM,respectively.(5)φisasurjectionfromΩCtoS,whichisalsocalledastructurefunction.Forastatevectorx∈ΩC,φ(x)isthestateofthesystem.Theleastandthegreatestelementsmeanthefullfailureandtheperfectfunctioningstates,respectively.TheordersonΩi(i∈C),Saredenotedbythecommonsymbol5,anda 119September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookModulesofMulti-StateCoherentSystems—OrderTheoreticalRelations105(2)Asystemφiscalledstrongrelevantwheneverycomponentisrele-vanttothesystem.Thestrongrelevantpropertyofthecomponentimeansthatforarbi-traryassignedtwodifferentstatesrandsofthesystem,thecomponenticanshiftthestatefromrtosonlybychangingthestatesofthecomponent,remainingtheothercomponents’statesunchanged.Thestrongrelevant,whichiscalledrelevantin[Ohi(2011)],isaconditionapparentlystrongerthanthatofDefinition6.6.Definition6.5.(StrongCoherentSystem)Asystemφiscalledstrongcoherent,whenφisincreasing,normalandstrongrelevant.Definition6.6.(RelevantProperty)(1)Thecomponenti∈Cissaidtoberelevantwhenthefollowingissatisfied.∀k,∀l∈Ωi(k̸=l),∃(ki,x),∃(li,x),φ(ki,x)̸=φ(li,x).(2)Asystemφiscalledrelevantwheneverycomponentisrelevant.Thecondition(1)ofDefinition6.6hasessentiallynopracticalrestrictiononasystem,sinceiftheconditiondoesnotholdforacomponenti∈C,wehave∃k,∃l∈Ωi(k̸=l),∀(ki,x),∀(li,x),φ(ki,x)=φ(li,x),whichmeansthatthestateskandlequivalentlycontributetothesystemstatesandthereforearenotnecessarilydefinedtobedifferentstates,inotherwords,wemaycombinethetwostateskandlintoonestate.Definition6.7.(CoherentSystem)Asystemφiscalledcoherent,whenφisincreasing,normalandrelevant.Seriesandparallelsystemsplayimportantrolesinthetheoryandprac-ticalsituationasseriesandparalleldecompositionofsystems,derivingstochasticboundsforreliability,andsoon.See[Ohi(2010)],[Ohi(2011)].Definition6.8.(SeriesandParallelSystems)Letφbeincreasing.(1)φiscalledaseriessystemwhenforeverys∈S,φ−1(s5)hastheleastelementwhichissimplywrittenasms.(2)φiscalledaparallelsystemwhenforeverys∈S,φ−1(5s)hasthegreatestelementwhichissimplywrittenasMs. 120September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book106ReliabilityModelingwithApplicationsProposition6.1.(1)Let(ΩC,S,φ)beastrongcoherentseriessystemandSisatotallyorderedset.Thenwehave∀s,∀t∈S(s 121September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookModulesofMulti-StateCoherentSystems—OrderTheoreticalRelations107Definition6.9.(Adefinitionofak-out-of-nsystem)Asystemφiscalledak-out-of-nsystem,whenforeachsubsetAofCsuchthat|A|=k,thereexistsaseriessystem(ΩA,φA,S)satisfying∀x∈Ω,φ(x)=infφ(xA).CAA⊂C,|A|=kDefinition6.10.(Adefinitionofaconsecutivek-out-of-nsystem)Asys-temφiscalledaconsecutivek-out-of-nsystem,whenforeachsubsetAofCsuchthatA={i,i+1,···,i+k−1},whichiscalledak-consecutivesubsetofΩ,thereexistsaseriessystem(ΩA,φA,S)satisfying∀x∈Ω,φ(x)=infφ(xA),CAA:ak-consecutivesubsetofCwhereifj∈Aisgreaterthann,jisconsideredtobej−n,thenthisdefini-tionisageneralizationofso-calledcircularconsecutivek-out-of-nsystem.Whenthestatespacesarebinarysets,thestructurefunctionsofthesedefinitionsareeasilyshowntobeusualbinarystatek-out-of-nandconsec-utivek-out-of-nsystems.Inthispaperwepresentonlydefinitionsofthesesystemsofwhichpreciseexaminationsasordertheoreticandprobabilisticexaminationsareremainedforfuturework.3ModulesFollowing[Ohi(2010)],weexaminetheconceptofmodulesbyusingtheφequivalentrelation=.φLetting(ΩC,S,φ)beasystem,wedefineapseudo-order5onaproductsetΩAofΩi(i∈A),whereAisanon-emptysubsetofC.Forx,y∈ΩAφdefx5y⇐⇒∀z∈ΩA′,φ(x,z)5φ(y,z),′φwhereA=C\A.Furthermoreanequivalentrelation=isdefinedonΩAasforx,y∈ΩA,φφφdefx=y⇐⇒∀z∈ΩA′,φ(x,z)=φ(y,z)⇐⇒x5yandy5x.φΩA|=isthequotientspaceofΩAwithrespecttotheequivalentrelationφφ=.EachelementofΩA|=iscalledanequivalentclass.Defininganorderφφ5onΩA|=asφφφdefD,E∈ΩA|=,D5E⇐⇒x5y(x∈D,y∈E), 122September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book108ReliabilityModelingwithApplicationsφφwehaveanorderedset(ΩA|=,5).Thisdefinitionisnotdependonthechoiceofxandy,sinceDandEareequivalentclasses.Forx∈ΩA,[x]istheequivalentclasstowhichxbelongs.WedefineaφφmappingφA:(ΩA|=,5)×ΩA′→Sasφdef([x],y)∈ΩA|=×ΩA′,φA([x],y)=φ(x,y),φφandχA:ΩA→(ΩA|=,5)asx∈ΩA,χA(x)=[x].Thesemappingsarerelatedwitheachotheras()′x∈Ω,φ(x)=φχ(xA),xA.(3)CAAProposition6.2.φ:increasing⇐⇒χA,φA:increasing.Proof.(onlyifpart)Letφ:ΩC→Sbeincreasing.Forx,y∈ΩA,x5y,wehave∀z∈ΩA′,(x,z)5(y,z)thenφ(x,z)5φ(y,z),whichmeansφ[x]5[y]i.e.χA(x)5χA(y).φ′′′′Forx,y∈ΩAsuchthat[x]5[y]andx,y∈ΩA′suchthatx5ywehavethefollowingchainofinequalities.φ([x],x′)=φ(x,x′)5φ(y,x′)5φ(y,y′)=φ([y],y′),AAφwherethefirstinequalityisfromx5y,andthesecondisfromtheincreas-ingpropertyofφ.′′′′(ifpart)Forx,y∈ΩA,x,y∈ΩA′suchthat(x,x)5(y,y),φ(x,x′)=φ(χ(x),x′)5φ(χ(y),y′)=φ(y,y′),AAAAsincex5yandtheincreasingpropertyofχAimplyχA(x)5χA(y),andfurthermorex′5y′andtheincreasingpropertyofφimplytheaboveAinequality. 123September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookModulesofMulti-StateCoherentSystems—OrderTheoreticalRelations109Proposition6.3.φ:relevant⇐⇒χA,φA:relevant.Proof.(onlyifpart)Supposei∈A.iisrelevanttoφ,then∀k,∀l∈Ωi(k̸=l),∃(·i,x),φ(ki,x)̸=φ(li,x).φ[][]Therefore(k,x)A̸=(l,x)A,i.e.,(k,x)A̸=(l,x)Aandtheniiii()()χk,xA\{i}̸=χl,xA\{i},AiAimeaningtherelevantpropertyofχA.Therelevantpropertyofφiseasilyproved.Supposei∈A′.A∀k,∀l∈Ωi(k̸=l),∃(·i,x),φ(ki,x)̸=φ(li,x).Thenwehave([]′)([]′)φxA,k,xA\{i}̸=φxA,l,xA\{i},AiAimeaningthatthecomponenti∈A′isrelevanttoφ.AφNextwesupposethatforx,y∈ΩA,[x]̸=[y]holds.Fromthedefinitionφoftheequivalentrelation=,φ(x,z)̸=φ(y,z)forsomez∈ΩA′.Notic-ingtherelation(3),wehaveφA([x],z)̸=φA([y],z).ThentherelevantpropertyofφAisproved.Theifpartisapparent,andthentheproofisomitted.Proposition6.4.Whenφisincreasing,wehavethefollowingrelationabouttheminimalelements.()()∀s∈S,x∈MIφ−1(s)=⇒xA∈MI[xA].Formaximalelements,wehaveasimilarrelation.Proof.Ifz5xAandz∈[xA],then()()()()′′′′φz,xA=φxA,xA=s,z,xA5xA,xA.()()′′Sincexisaminimalelementofφ−1(s),z,xA=xA,xAandthusz=xAfollows.Proposition6.5.Whenφisincreasing,foreverys∈Sandx∈ΩAwehavethefollowingrelation.()()([x],y)∈MIφ−1(s)andx∈MI([x])=⇒(x,y)∈MIφ−1(s).AmmFormaximalelements,wehaveasimilarrelation. 124September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book110ReliabilityModelingwithApplicationsProof.Wesupposeu∈ΩA,v∈ΩA′,(u,v)5(xm,y),φ(u,v)=s,andnotice[xm]=[x].φSinceu5xm,[u]5[xm]andthen([u],v)5([xm],y).Ontheotherhandφ(u,v)=φA([u],v)=s,then([u],v)=([xm],y),because([x],y)=−1([xm],y)isaminimalelementofφA(s).Furthermore[u]=[xm]whichmeansu∈[xm],xmisaminimalelementof[x],andu5xm,thenwehaveu=xm.Proposition6.6.φ:increasing,normal=⇒φA:normal.Proof.Wesuppose()−1s∈S,t∈S,s 125September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookModulesofMulti-StateCoherentSystems—OrderTheoreticalRelations111φ[]Since[y]5xAandχisnormal,wehaveA∃y∈MI([y]),y5xA,mmandthenfinally()′(y,z)5xA,xA,φ(y,z)=φ([y],z)=φ([y],z)=s,mmAmAwhichmeansφtobenormal.Proposition6.8.WhenφisincreasingandχAisnormal,thenwehavethefollowingrelationaboutminimalelements.()(′)()x∈MIφ−1(s)=⇒[xA],xA∈MIφ−1(s)AThesimilarrelationholdsformaximalelements.()([]′)Proof.Letx∈MIφ−1(s).AninclusionrelationxA,xA∈([])()−1AA′AA′φA(s)isclearfromφAx,x=φx,x=s.Wesuppose([y],z)tobe([]′)([]′)φ([y],z)=s,([y],z)5xA,xA,([y],z)̸=xA,xA,Aφnoticingthat[y]5[xA]holds.′′If[y]=[xA],thenz5xAandz̸=xA,therefore()(′)()(′)()xA,z5xA,xA,xA,z̸=xA,xA,φxA,z=s,()contradictingtox∈MIφ−1(s).[][]([])Hencewehave[y]5xAand[y]̸=xA.SincexA∈MIxAbyProposition6.4,thereexistsaminimalelementymof[y]fromthenormalpropertyofχAandthuswehave()()′′φ(y,z)=φ(y,z)=φ([y],z),(y,z)5xA,xA,(y,z)̸=xA,xA,mAmm()()′whichcontradictstox=xA,xA∈MIφ−1(s).Fromtheabovepropositions,increasing,normalandrelevantpropertiesofφdeterminetheincreasingandrelevantpropertiesofφAandχA,andfurthermorethenormalpropertyofφA,butnotthenormalpropertyof()φχA.ThenA⊂Cisamoduleornot,inotherwords,ΩA,ΩA|=,χAisacoherentsystemornot,accordingtoχAisnormalornot,respectively. 126September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book112ReliabilityModelingwithApplicationsSummarizedrelationsareshowninthefollowing.φ:increasing⇐⇒φA,χA:increasing,φ:relevant⇐⇒φA,χA:relevant,φ:increasingandnormal=⇒φA:normal,φ:normal⇐=φA,χA:normal.Whenφisincreasing,x∈MI(φ−1(s))=⇒xa∈MI([xa]),([x],y)∈MI(φ−1(s))andx∈MI([x])=⇒(x,y)∈MI(φ−1(s)).AmmWhenφisincreasingandχAisnormal,()−1aA′−1x∈MI(φ(s))=⇒[x],x∈MI(φ(s)).AHere,remindingthatasystemiscalledcoherentinthispaperwhenthesystemisincreasing,relevantandnormalfromDefinition6.7,wehavethefollowingtheorem.Theorem6.1.When(ΩC,S,φ)isacoherentsystem,A⊂CisamoduleifandonlyifχAisnormal.FromPropositions6.4,6.5and6.8,wehavethenexttheoremaboutmaximalandminimalelements.Theorem6.2.Letφbeacoherentsystem,andA⊂Cisassumedtobeamodule.Forx∈ΩA,y∈ΩA′,wehavethefollowingequivalentrelation;()()(x,y)∈MIφ−1(s)⇐⇒([x],y)∈MIφ−1(s)andx∈MI([x]).ASimilarequivalentrelationholdsformaximalelements.Remark6.1.Normalpropertyseemstoplayanimportantrolefordefiningmodules,sincefromtheabovepropositions,whenwedonotassumethenormalpropertyonthedefinitionofmodules,everysubsetAofCmaybeamodule,whichmeansnobrakestodefiningthemodules.Itiswellknownthateverysubsetofcomponentsisamoduleinthecaseofbinarystateseriesandparallelsystems.Wehaveasimilarconclusionforthecaseofmulti-statesystems.Hereweprovethepropositionforstrongcoherentsystemswhenthestatespacesaretotallyorderedsets.Proposition6.9.Supposeasystem(ΩC,S,φ)isastrongseries(parallel)systemandthestatespacesaretotallyorderedsets.TheneverysubsetAofcomponentsofthesystemisamodule. 127September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookModulesofMulti-StateCoherentSystems—OrderTheoreticalRelations113Proof.Withoutlossofgenerality,wemayassumeS={0,1,···,M},sinceSisassumedtobeafinitetotallyorderedset.FromProposition6.1(1),m0< 128September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book114ReliabilityModelingwithApplications4ConcludingRemarksInthispaperwepresentedadefinitionofmodulesandshowedthatthenormalpropertyplaysanimportantroleforthedefinition,sinceifwedonotassumethenormalpropertyonthedefinitionofsystems,anysubsetofthecomponentscouldbemoduleinthecontextofpartiallyorderedsetsandnobrakeisappliedtodefiningmodules,inotherwords,thedefinitionofmoduleswithouttheassumptionofnormalpropertyismeaningless.Fur-thermoreweshowedthatanysubsetofcomponentsconsistingacoherentseriesorparallelsystemisamoduleinthemulti-statecaseofwhichstatespacesaretotallyorderedsets,followingthedefinitionofseriesandparallelsystemsof[Ohi(2010)].Ontheotherhand,weleftsomeproblemsforfutureworks.Inthecaseofbinarystatecase,itiswellknownthatanyk-out-of-nsystem(25k5n−1)hasnomodule.Itisleftanopenproblemtoprovethesimilarassertionforthemulti-statecase.Thesecondproblem,whichismoreimportant,istogivestochasticboundsforsystemsthroughmodules,alsoanopenproblem.ReferencesBarlow,R.E.andProschan,F.(1975).StatisticalTheoryofReliabilityofLifeTesting,Holt,RinehartandWinston,NewYork.Barlow,R.E.andAlexander,S.Wu(1978).Coherentsystemswithmultistatecomponents,MathematicsofOperationsResearch,3,pp.275–281.Birnbaum,Z.W.andEsary,J.D.(1965).Modulesofcoherentbinarysystems,SIAMJ.Appl.Math.,13,pp.444–462.Birnbaum,Z.W.,Esary,J.D.andSaunder,S.C.(1961).Multi-componentsys-temsandstructuresandtheirreliability,Technometrics,3,pp.55–77.El-Neweihi,E.,Proschan,F.andSethuraman,J.(1978).Multistatecoherentsystems,J.Appl.Probability,15,pp.675–688.Esary,J.D.andProschan,F.(1963).Coherentstructuresofnon-identicalcom-ponents,Technometrics,5,pp.191–209.Esary,J.D.,Marshall,A.W.andProschan,F.(1970).Somereliabilityapplica-tionofhazardtransform,SIAMJ.Appl.Math.,18,pp.331–359.Hirsch,W.M.,Meisner,M.andBoll,C.(1968).Cannibalizationinmulticompo-nentsystemsandthetheoryofreliability,NavalRes.Logist.Quart.,15,pp.331–359.Hochberg(1973).Generalizedmultistatesystemsundercannibalization,NavalRes.Logistic.Quart.,20,pp.585–605.Huang,J.,Zuo,M.J.andFang,Z.(2003).Multi-stateconsecutive-k-out-of-nsystems,IIETransactions,35,pp.527–534.Mine,H.(1959).Reliabilityofphysicalsystem,IRE,CT-6SpecialSupplement,pp.138–151. 129September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookModulesofMulti-StateCoherentSystems—OrderTheoreticalRelations115Ohi,F.andNishida,T.(1983).Generalizedmultistatecoherentsystems,J.JapanStatist.Soc.,13,pp.165–181.Ohi,F.andNishida,T.(1984).OnMultistateCoherentSystems,IEEETrans-actionsonReliability,R-33,pp.284–288.Ohi,F.andNishida,T.(1984).MultistateSystemsinReliabilityTheory,Stochas-ticModelsinReliabilityTheory,LectureNotesinEconomicsandMathe-maticalSystems235,Springer-Verlag,pp.12–22.Ohi,F.(2010).MultistateCoherentSystems,in“StochasticReliabilityModeling,OptimizationandApplications”,editedbyS.NakamuraandT.Nakagawa,WorldScience,pp.3–34.Ohi,F.(2011).LatticeSetTheoreticTreatmentofMulti-stateCoherentSystems,ProceedingsofThe7thInternationalConferenceon“MathematicalMethodinReliability”:Theory.Methods.Applications,editedbyLirongCui&XianZhao,pp.383–389.Ohi,F.(2012).Multi-StateCoherentSystemsandModules–BasicProperties–,The5thAsia-PasificInternationalSymposiumonAdvancedReliabilityandMaintenanceModeling,inAdvancedreliabilityandmaintenancemodelingv,BasisofReliabilityAnalysis,Nanjing,China,1-3November2012/12/02,editedbyHisashiYamamoto,ChunhuaQian,LirongCuiandTakashiDohi,McGrowHillEducation,pp.374–381.Shinmori,S.,Ohi,F.,Hagihara,H.andNishida,T.(1989).ModulesforTwoClassesofMulti-StateSystems,TheTransactionsoftheIEICE,E72,pp.600–608.Yu,K.,Koren,I.andGuo,Y.(1994).GeneralizedMultistateMonotoneCoherentSystems,IEEETransactionsonReliability,43,pp.242–250. 130May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank 131September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookChapter7CalculationAlgorithmsfortheSystemStateDistributionsofMulti-Statek-out-of-nSystemsHisashiYamamoto1andTomoakiAkiba21FacultyofSystemDesign,TokyoMetoropolitanUniversity,6-6Asahigaoka,Hino,Tokyo191-0065,Japan2FacultyofSocialSystemScience,ChibaInstituteofTechnology,2-17-1,Narashino,Chiba275-0016,Japan1IntroductionIntraditionalreliabilitytheory,boththesystemanditscomponentsareconsideredtotakeonlytwopossiblestates:workingorfailed.Inthebinarycontext,asystemwithncomponentsinsequenceiscalledthek-out-of-n:F(G)systemifthesystemfails(works)wheneveratleastkcomponentsinthesystemwork(fail).Whenk=n,k-out-of-n:F(G)systemisparallel(series)system.Inmanypracticalsituations,thestatesofthesystemsandtheircom-ponentsareconsideredtotakemorethantwodifferentlevels,rangingfromperfectlyworkingtocompletelyfailed.Manyresearchershaveextendedbinarysystemofk-out-of-n:F(G)system,forexample,see,[BarlowandProchan(1975)],[Kolowrocki(2004)],[Changetal.(2000)]and[KuoandZuo(2003)].Inthemulti-statek-out-of-n:F(G)system,boththesystemandthecomponentsareallowedtobeinM+1possiblestates,0,1,2,···,M,whereMisapositiveintegerwhichrepresentsthesystemorcomponentinperfectlyworkingstate,whilezerorepresentscompletelyfailurestate.Themulti-statek-out-of-nsystemreliabilitymodelprovidesmoreflexibil-ityforthemodelingofequipmentconditions.Somepapersprovidedwith117 132September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book118ReliabilityModelingwithApplicationsthealgorithmsforcomputingsystemstatedistributionofmulti-statesys-tem.Researchershaveextendedbinaryk-out-of-nsystemtomulti-statek-out-of-nsystem,forexample,see,[BarlowandWu(1978)],[El-Neweihietal.(1978)],[Griffith(1980)],[KossowandPreuss(1995)],[MalinowskiandPreuss(1995,1996)],[ZuoandLiang(1994)],[Koutras(1997)],[HaimandPorat(1991)],[Huangetal.(2000)],[Zuoetal.(2003,2006)],[Amarietal.(2009)]and[Tianetal.(2009)].[Huangetal.(2000)]proposedthedefinitionofgeneralizedmulti-statek-out-of-n:Gsystemandaneffi-cientalgorithmforevaluatingthesystemstatedistributionsineachcaseofdecreasing,increasingandconstantmulti-statek-out-of-n:Gsystems.[Yamamotoetal.(2006)]proposedanefficientalgorithmforevaluatingthesystemstatedistributionofthegeneralizedmulti-statek-out-of-n:Gsysteminnon-i.i.d.case.Ontheotherhand,[Zuoetal.(2003)]proposedadefinitionofthegeneralizedmulti-statek-out-of-n:Fsystemandaneffi-cientalgorithmforevaluatingthesystemstatedistributionofadecreasingmulti-statek-out-of-n:Fsystem.[Zuoetal.(2006)]describesmulti-statek-out-of-n:Fsystembecomesmulti-state(n−k+1)-out-of-n:Gsystem.Therefore,systemstatedistributionofgeneralizedmulti-statek-out-of-n:Fsystemcanbeevaluatedbyusingalgorithmsformulti-statek-out-of-n:Gsystem,forexample,[Yamamotoetal.(2006)]’salgorithm.However,theiralgorithmsarenotsoefficientifklaresmall(l=1,2,···,M)foramulti-statemulti-statek-out-of-n:Fsystem.Accordingly,[Yamamotoetal.(2011)]proposedafasteralgorithmusingideaofvirtualcompo-nentstates,whichisefficienteveninnon-i.i.d.case,forevaluatingthesystemstatedistributionofthegeneralizedmulti-statek-out-of-n:Fsystembyextending[Yamamotoetal.(2009)]’salgorithmproposedforevaluat-ingthesystemstatedistributionofageneralizedmulti-statek-out-of-n:Fsystem.Inthisarticle,weshowefficientalgorithmsforcomputingsystemstatedistributionofmulti-statek-out-of-n:Fsystemandmulti-statek-out-of-n:Gsystembyusingrecursivealgorithmsandtechniquesofvirtualcomponentstateswhichenableustoreducetheconsiderednumberofstates.Wepresenttheorderofcomputingtimeandmemorycapacityofproposedalgorithms.Wealsoperformnumericalexperiments.Theresultsshowthattheproposedalgorithmsaremoreefficientthantheexistingalgorithmsforevaluatingthesystemstatedistributionofmulti-statek-out-of-n:Fsystemandmulti-statek-out-of-n:Gsystem. 133September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookCalculationAlgorithmsfortheSystemStateDistributions1192DefinitionfortheMulti-statek-out-of-nSystemsLetubevectorofcomponentstatesu=(u1,u2,···,un)whereuiisstateofcomponenti,ui∈{0,1,···,M},fori=1,2,···,n.And,ϕ(u)definesystemstructurefunctionrepresentingthestateofthesystem,ϕ(u)∈{0,1,···,M}.And0meansn-dimensionalzerovector.First,weshowthedefinitionofgeneralizedmulti-statek-out-of-n:Fsys-temasfollows.Definition[Zuoetal.(2003)]:ϕ(u) 134September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book120ReliabilityModelingwithApplicationsstateloraboveforalll.Ann-componentsystemwithsuchapropertyiscalledamulti-statek-out-of-n:Gsystem.Theconditioninthisdefinitioncanalsobephrasedasfollows:atleastkjcomponentsareinstatesbelowj,oratleastkj+1componentsareinstatesbelowj+1,or···,oratleastkMcomponentsareinstatesbelowM.Next,weassumedthefollowingthroughoutthisarticle.Assumption:(1)Multi-statek-out-of-nsystemsaremulti-statemonotone[Griffith(1980)].Thatis,systemstructurefunctionϕ(u)satisfies,i.ϕ(u)isincreasingfunctionofu≥0and,ii.uiorequivalently,ϕ(u)=jforj=0,1,···,M.[Huangetal.(2000)]and[Zuoetal.(2003)]consideredspecialcasesofthisdefinitionofmulti-statek-out-of-n:F(G)system.Whenk1≥k2≥···≥kM(k1 135September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookCalculationAlgorithmsfortheSystemStateDistributions1213TheoremandAlgorithmfortheSystemStateDistribu-tionsoftheGeneralizedMulti-Statek-out-of-n:FSystemInthissection,weshowanalgorithmforevaluatingthesystemstatedistri-butionofthegeneralizedmulti-statek-out-of-n:Fsystem[Yamamotoetal.(2011)]byunifying[Tianetal.(2009)]’sand[Yamamotoetal.(2009)]’sideas.Now,wedefinethefollowingnotations.Fori=1,2,···,n,andj=1,2,···,M,pij:probabilitythatcomponentiisinstatej.F(j)(i;k,P):probabilitythatstateofthemulti-statek-out-of-i:Fsystemisbelowjwherek=(k1,k2,···,kM)andp1,0p1,1···p1,Mp2,0p2,1···p2,MP≡.............pn,0pn,1···pn,MAnd,thefollowingareassumedthroughoutthisarticle.Assumption:(2)Stateofcomponentoccurstomutuallystatisticallyindependent.TheimportantpointofproposedalgorithmistoreducetheconsiderednumberofstatesbyusingCorollary2whichisdescribedlater.Forthis,weconsidervirtualcomponentstates,eachofwhichconsistsofsomeactualcomponentstates.Weexpressthevirtualcomponentstatesas¯0,¯1,¯2,···,m¯where¯0<¯1<¯2<··· 136September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book122ReliabilityModelingwithApplicationsFig.1AnexamplefortherelationofvirtualcomponentstateswitheventsS(i,l;xl),whenm=3,(x1,x2,x3)=(2,4,6)andi=8ρi,j:probabilitythatcomponentiisinvirtualstatesj.P(¯m):n×(m+1)matrixofthecomponentstateprobability,thatis,ρ1,¯0ρ1,¯1···ρ1,m¯ρ2,¯0ρ2,¯1···ρ2,m¯P(¯m)≡.............ρn,¯0ρn,¯1···ρn,m¯Thenρi,jcanbeexpressedasjl∑+1−1ρi,l=pi,tt=jlforl=¯0,¯1,¯2,···,m¯,wherej¯0≡0andjm¯+1≡M+1.Wealsodefinenotationsasfollows.Fori=0,1,···,n,xl=0,1,···,kl,l=¯1,¯2,···,m¯andm=1,2,···,M,S(i,l;xl):theeventthatatleastxlcomponentsin(virtualoractual)statebelowloccurfromcomponent1toiwheni=1,2,···,n;nulleventwhenxl>i;wholeeventwheni=0.∩m¯Q(i,(x(¯m)¯1,x¯2,···,xm¯),P):Pr{S(i,l,xl)}whenρi,jistheprobabilityl=¯1thatcomponentiisinthevirtualcomponentstatej.Figure1showstherelationbetweenthevirtualcomponentstatesandS(i,l;xl)byusingasimpleexample.WearenowreadytopresentthefollowingTheorem. 137September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookCalculationAlgorithmsfortheSystemStateDistributions123Theorem7.1.1)Fori=1,2,···,nandm=M−j+1,F(j)(i;k,P)=Q(i,(k,k,···,k),P(¯m))(1)jj+1Mwhere∑j−1p1,lp1,j···p1,Ml=0∑j−1p2,lp2,j···p2,MP(¯m)≡.l=0............∑j−1pn,lpn,j···pn,Ml=02)Fori=0,1,···,nandl=¯1,¯2,···,m¯,Q(i,(x(¯m)¯1,x¯2,···,xm¯),P)=0if∃l:x>i,l1if(x¯1,x¯2,···,xm¯)=0,∑m¯ρi,hQ(i−1,(x¯1,x¯2,···,xh,h=¯0max{xh+1−1,0},···,max{x−1,0}),P(¯m))m∀ifxl≤iforlandi>0.(2)Theorem7.1canbeprovenwithasimilarmannertoTheorem1in[Yamamotoetal.(2009)].FromTheorem7.1,wecangetthefollowingCorollaryforthesystemstatedistributionofabinaryk-out-of-n:Fsystem.Corollary7.1.Fori=0,1,···,n,0ifx¯1>i,Q(i,(x(¯1)¯1),P)=1ifx¯1=0,ρi,¯0Q(i−1,(max{x1−1,0}),P(¯1))ifx¯1≤iandi>0.(3)Corollary7.1isthesameastraditionaltheoremofreliabilityofabinaryk-out-of-n:Fsystem.(see,forexample,[Rushdi(1986)]etc..) 138September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book124ReliabilityModelingwithApplicationsNext,fromthedefinitionoftheeventS(i,l;xl),wecangetthefollowingLemma.Lemma7.1.Fori=0,1,···,n,ifxt¯≥max{xl}thenwecangetthel=t+1,···,s¯followingrelationoftheeventSF(i,l;x)s.l∩s¯S(i,l;xl)=S(i,t¯;xt¯).(4)l=t¯TheproofofLemma7.1isprovidedinappendixat[Yamamotoetal.(2009)].FromLemma7.1,wecangetthefollowingTheorem.Theorem7.2.Wetakexj(l)∈{x¯1,x¯2,···,xm¯}forl=1,2.1)Ifxj(2)existssuchthatxj(1)≥max{xl}andxj(1) 139September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookCalculationAlgorithmsfortheSystemStateDistributions125fori=0,1,···,n,where∑m¯ρ1,¯0ρ1,¯1···ρ1,ll=j(1)∑m¯ρ2,¯0ρ2,¯1···ρ2,lP(j(1))=.l=j(1)...∑m¯ρn,¯0ρn,¯1···ρn,ll=j(1)TheproofofTheorem7.2isprovidedinappendixat[Yamamotoetal.(2011)].FromTheorem7.2,wecangetthefollowingCorollary.Corollary7.2.Wetake{x¯1,xj(1),···,xj(r)}⊆{x¯1,x¯2,···,xm¯}.Ifxj(t)≥max{xl},xj(t) 140September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book126ReliabilityModelingwithApplications(3,2,1,5,4),wecangetQ(i,(3,5),P(¯2))=Q(i,(3,2,1,5,4),P(¯5))fromx¯1>max{x¯2,x¯3},x¯4>x¯5andx¯1 141September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookCalculationAlgorithmsfortheSystemStateDistributions127ρ1,¯0ρ1,¯1ρ1,¯2p1,0p1,1p1,2ρ2,¯0ρ2,¯1ρ2,¯2p2,0p2,1p2,2P(¯2)==.......ρn,¯0ρn,¯1ρn,¯2pn,0pn,1pn,2FromEquation2,{(¯2)0if(x1,x2)̸=(0,0),Q(i,(x1,x2),P)=1if(x1,x2)=(0,0),fori=0andQ(i,(x,x),P(¯2))=120ifx1>iorx2>i,1ifx1=0andx2=0,pQ(i−1,(x−1,x−1),P(¯2))i,012(¯2)+pi,1Q(i−1,(x1,x2−1),P)+pQ(i−1,(x,x),P(¯2))i,212ifi≥x2>x1>0,pQ(i−1,(0,x−1),P(¯2))i,02+pi,1Q(i−1,(0,x2−1),P(¯2))+pQ(i−1,(0,x),P(¯2))i,22ifx1=0andi≥x2>0,pQ(i−1,(x−1),P(¯1))i,01(¯1)+(pi,1+pi,2)Q(i−1,(x1),P)ifi≥x1≥x2>0,fori=1,2,···,nandxl=0,1,···,kl(l=1,2).UsingTheorem7.1andCorollaries7.1and7.2,weobtainthefollowingalgorithmstepsforcomputingF(j)(n;k,P)forj+1,2,···,M.STEP0:(Settinginitialvalue)Seti=0,m=M−j+1and{(¯m)0if(x¯1,x¯2,···,xm¯)̸=0,Q(i,(x¯1,x¯2,···,xm¯),P)=1if(x¯1,x¯2,···,xm¯)=0,fromEquation2.STEP1:Seti=i+1.STEP2:Enumerateall(x¯1,x¯2,···,xm¯)∈A,whereA≡{(x¯1,x¯2,···,xm¯)|x¯l=0,1,···,kl+j−1(l=1,2,···,m)}. 142September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book128ReliabilityModelingwithApplicationsSTEP3:Pickup(x¯1,x¯2,···,xm¯)∈A.Forthe(x¯1,x¯2,···,xm¯),sup-poseweobtain,x,···,x),P(r+1))=Q(i,(x(¯m)Q(i,(x¯1,xj(1)j(2)j(r)¯1,x¯2,···,xm¯),P)fromCorollary7.2.STEP4:ObtainQ(i,(x,x,···,x),P(r+1))usingTheorem¯1,xj(1)j(2)j(r)7.1andmemorizeQ(i,(x(r+1)¯1,xj(1),xj(2),···,xj(r)),P).GotoSTEP5ifall(x¯1,x¯2,···,xm¯)’s∈Aarepickedup.Whenoth-erwise,gotoSTEP3.STEP5:GotoSTEP1wheni 143September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookCalculationAlgorithmsfortheSystemStateDistributions129First,weproposethefollowingTheorem.Somenotationsaresameastheprevioussection.Theorem7.3.1)Fori=1,2,···,nandj=1,2,···,M,R(j)(i;k,P)=1−F(i,(k,k,···,k),P(M−j+1))(8)jj+1Mwhere∑j−1p1,lp1,j···p1,Ml=0∑j−1p2,lp2,j···p2,MP(M−j+1)≡.l=0............∑j−1pn,lpn,j···pn,Ml=02)Fori=0,1,···,n,l=¯1,¯2,···,m¯andm=1,2,···,M,F(i,(x(¯m)¯1,x¯2,···,xm¯),P)=1if∀l:x>i,l0ifmin{xl}=0,l∑m¯ρF(i−1,(y(¯m)i,h¯1,···,yh,yh+1,···,ym¯),P)otherwise,h=¯1(9)where0ifxl≤0,yl=xl−1ifxl>0andl≤h,xlifxl>0andl>h.3)Fori=0,1,···,nandm=1,F(i,(x(¯1)¯1),P)=1ifx¯1>i,0ifx¯1=0,(10)ρF(i−1,(max{x¯−1,0}),P(¯1))i,¯01(¯1)+ρi,¯1F(i−1,(x¯1),P)ifx¯1≤i,i>0. 144September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book130ReliabilityModelingwithApplicationsTheorem7.3canbeprovenwithasimilarmannertoTheorem7.1.Equa-tion10meansboundaryconditionwhenthenumbermofvirtualstatesofcomponentsis1.Then,wecangetthesystemstatedistributionbyusingequationforthereliabilityofabinaryk-out-of-n:Gsystem.Inaddition,fromthedefinitionoftheeventSF(i,l;x),wecangetthelfollowingLemma.Lemma7.2.Fori=0,1,···,n,s=1,2,···,mandt=1,2,···,m,ifxs¯≤min{xl}thenwecangetthefollowingrelationoftheeventl=s+1,···,t¯SF(i,l;x)s.lt¯∩SF(i,l,x)=SF(i,s¯;x).(11)ls¯l=¯sFromLemma7.2,wecangetthefollowingLemma.Lemma7.3.Wetakexj(2)∈{x¯1,···,xm¯}.1)Ifxexistssuchthatx 145September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookCalculationAlgorithmsfortheSystemStateDistributions1312)Ifxj(2) 146September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book132ReliabilityModelingwithApplicationsTheprobabilityF(i,(k,k,···,k),P(M−j+1))canbegivenbythejj+1MfollowingTheorem.Theorem7.4.Now,wetake{kj(1),kj(2),···,kj(r)}⊆{k1,k2,···,kM}wherej≤j(1) 147September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookCalculationAlgorithmsfortheSystemStateDistributions133STEP1:Seti=i+1.STEP2:Setm=M,andenumeratecombinationsforallelementxl(l=¯1,¯2,···,m¯,xl∈{0,,···,max{xl}})STEP3:Reducedtheelementsforallxls(l=¯1,¯2,···,m¯)byusingLemma7.3.STEP4:Forallxls(l=¯1,¯2,···,m¯)obtainedintheSTEP3,obtainthevalueofyls(l=¯1,¯2,···,m¯)forallhbyusingtheEquation11or10inTheorem7.3.Next,wereducedtheelementsforallyls(l=¯1,¯2,···,m¯)byusingLemma7.3.STEP5:ObtainF(i−1,(y(¯m)¯1,···,ym¯),P)fromthememorybyus-ingylsthatthesewerecalculatedinSTEP4.Next,obtainF(i−1,(x(¯m)¯1,···,xm¯),P)byusingEquation11or10inTheorem7.3,andmemorizetheresultofF(i−1,(x(¯m)¯1,···,xm¯),P).STEP6:GotoSTEP7ifwefinishedenumeratingcombinationsforallelementxls,andgotoSTEP2otherwise.STEP7:GotoSTEP1ifi 148September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book134ReliabilityModelingwithApplicationsetal.(2009)]’salgorithm.However,astheproposedalgorithmreducethenumberofcomponentstatesbyusingCorollary7.2,thecomputingtimeseemstobelessthanorequalto[Yamamotoetal.(2009)]’salgorithm.∏MNext,theorderoftherequiredmemorysizeisO(nkl)byusingl=1[Yamamotoetal.(2009)]’salgorithm.Thatis,theorderisalsoexponentialofM,butdoesnotdependonn.Ontheotherhand,thealgorithmfor∏McomputingQ(i,(x(¯m)2¯1,x¯2,···,xm¯),P)needsthemaximumof2(nkl)l=1∏Mentries,becauseweneedklentriesforeveniandoddi,respectively.l=1∏MTherefore,theorderoftherequiredmemorysizeisO((k)2)forthell=1proposedalgorithm.Formulti-statek-out-of-n:Gsystem,weevaluatetheordersofcomput-ingtimeandmemorysizefortheproposedalgorithminsection4.Theproposedalgorithmusesthesimilarrecursiveequationto[Yamamotoetal.(2006)]’salgorithm.InordertocomputeF(i,(x¯1,x(¯m)¯2,···,xm¯),P)’sforeachiandallxl’ssuchthatxltakesklfor∏Malll(l=¯1,¯2,···,m¯),wemustuseEquation11amaximumofkll=1times.Therefore,theorderofobtainingallF(i,(x(¯m)¯1,x¯2,···,xm¯),P)sfor∏Mj=1,2,···,MisO(Mnkl)byLemma7.3.Thisisthesameorderasl=1[Yamamotoetal.(2006)]’salgorithm.However,ourproposedalgo-rithmcanreducemorecomputingtimebyTheorem7.4forredundantx’s.Furthermore,allF(i,(x(¯m)l¯1,x¯2,···,xm¯),P)sforj=2,3,···,Mcanbeobtainedwhenj=1,becauseF(i,(x(¯m)¯1,x¯2,···,xm¯),P)sforj=2,3,···,Maresameforeachj.Therefore,theorderofobtainingF(i,(x(¯m)¯1,x¯2,···,xm¯),P)forallj=∏M1,2,···,MisO(nkl)byproposedalgorithm.l=1Accordingly,weestimatethattheproposedalgorithmcangetfasterorequaltocomputingtimebyusing[Yamamotoetal.(2006)]’salgorithm.Similarly,theorderofthememoryrequiredbytheproposedalgorithmis∏MO((k)2).ll=1 149September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookCalculationAlgorithmsfortheSystemStateDistributions135Theseresultsindicatethat(1)[Yamamotoetal.(2009)]and[Yamamotoetal.(2006)]’salgorithmsareadvantageousoverproposedalgorithmwithrespecttomemorysize,(2)proposedalgorithmsareadvantageousover[Yamamotoetal.(2009)]and[Yamamotoetal.(2006)]’salgorithmswithrespecttocomputingtime.5.2TheResultofNumericalExperimentsThoughwediscussedtheordersofcomputingtimeandmemorysize,theirordersareclose.So,weperformedanumericalexperimentinordertocom-pareproposedalgorithmswith[Yamamotoetal.(2006,2009)]’salgorithmsbyactualcomputingtimes.First,weevaluatedsystemstatedistributionsofincreasingfour-statek-out-of-n:Fsystemswithk1=4,k2=5,k3=6,k4=7.TheresultsareshowninTable1.Theproportionmeanstheratioofthecomputingtimeoftheproposedalgorithmto[Yamamotoetal.(2009)]’salgorithm.AlltheexperimentswereexecutedusingaPentium4(2.8GHz)computerwith1.0GBytesofRAM,MS-WindowsXP,VisualC++.NET2003andClanguageprogramming.InTable1,computingtimesaretheaveragesoffivetrialsforeachn.Fromtheresultsofthenumericalexperiments,weseethesetwoalgorithmsareefficientforevaluatingthesystemstatedistributionsofTable1Computingtimeforthesystemstatedistributionofincreasingfour-statek-out-of-n:Fsystems(sec.)nYamamotoetal.’sProposedProportionalgorithm(2009)algorithm100.1090.01513.8%200.2650.0166.0%300.3980.04511.3%400.5200.05911.3%500.7180.07911.0%1001.7900.1367.6%1502.3000.26511.5%2003.0000.36112.0%3004.9800.63112.7%5007.5000.89111.9%70013.2001.2679.6%100017.9001.7359.7% 150September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book136ReliabilityModelingwithApplicationsthegeneralizedmulti-statek-out-of-n:Fsystems,includingnotonlythedecreasing,increasing,andconstanttypes,butalsoothernon-monotonictypes.Table1showsthatthecomputingtimesoftheproposedalgorithmarelessthanthe[Yamamotoetal.(2009)]’salgorithm,especiallywhenthenumberofcomponentsnislarge.Similarly,weevaluatedcomputingtimesforfour-statek-out-of-n:Gsys-tems.AlltheexperimentswereexecutedusingaPentiumM1.3GHzCPU,768MBmemory,MicrosoftWindows2000,VisualC++.NET2003andClanguageprogramming.Thoughthenumericalexperiments,wecomparedcomputingtimesoftheproposedalgorithmwiththoseof[Yamamotoetal.(2006)]’salgorithm.Wecalculatedthesystemstatedistributionofadecreasing,increasingandgeneralizedfour-statek-out-of-n:GsystemasshowninTable2,whichareadverseconditionsfortheproposedalgorithm.InTable2,theresultsofcomputingtimesaretheaveragesfromfivetrialsforeachnvalue.Fromresultsofthenumericalexperiments,thesetwoalgorithmsareefficientformulti-statek-out-of-n:Gsystemwhenthenumberofnislarge.And,Table2showsthattheproposedalgorithmtakeslesscomputingtimethanthe[Yamamotoetal.(2006)]’salgorithm,especiallywhenthenumberofcomponentsnislarge.Frombothresultsofordersandnumericalexamples,weseethattheproposedalgorithmsareefficientforcomputationtimebyreducingredun-dantxl’s,andtheseareefficientforevaluatingthesystemstatedistributionforlargen.Table2Computingtimeforthesystemstatedistributionofincreasingfour-statek-out-of-n:Gsystems(sec.)k1=7,k2=6,k1=4,k2=5,k1=5,k2=4,k3=5,k4=4k3=6,k4=7k3=6,k4=7nProposedYamamotoProposedYamamotoProposedYamamotoAlgorithmetal.(2006)Algorithmetal.(2006)Algorithmetal.(2006)500.1400.1640.1600.1960.1620.1921000.2740.3340.3160.3950.3140.4071500.4100.5070.4750.5910.4750.5952000.5440.6770.6290.7910.6310.7953000.8211.0150.9491.1920.9451.2025001.3861.6981.5901.9991.5802.0117001.9082.3832.2062.8022.2092.8089002.4563.0662.8383.6012.8403.61110002.7153.4113.1534.0043.1554.014 151September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookCalculationAlgorithmsfortheSystemStateDistributions137However,theorderofthememoryshowstheproposedalgorithmsre-quirestoomuchmemorycapacityforlargeM.6ConclusionsInthisarticle,wediscussedefficientrecursivealgorithmforevaluatingthesystemstatedistributionofgeneralizedmulti-statek-out-of-n:F(G)sys-teminthenon-i.i.d.case.Weproposedalgorithmsareusingtheideaforreducingthenumberoftheconsideredcomponentstates.Wecomparedtheordersofcomputingtimeandmemorysizerequirementsofproposedalgorithmswithotheralgorithms.Thoughtheirordersareclose,fromtheresultsofnumericalexperiments,wecanconcludethattheproposedalgorithmscancalculatethesystemstatedistributionsofgeneralizedmulti-statek-out-of-n:F(G)systemsfasterthanotheralgorithmsinmostcases,especiallyforlargen,withintherangeofourexperiments.Suchanadvantageissignificantespeciallywhennislarge.However,thisalgorithmrequirestoomuchmemorycapacitytoevaluatethesystemstatedistributions,whenMislarge.Forsuchacase,wecanrecommendtouseotheralgorithms.AcknowledgmentThisworkwaspartiallysupportedbyGrantNo.23510204,Grant-in-AidforScientificResearch(c)fromJSPS(2011-).TheauthorsthanktheJSPSfortheirsupport.ReferencesAmari,S.V.,Zuo,M.J.andDill,G.(2009).AFastandRobustReliabilityEval-uationAlgorithmforGeneralizedMulti-Statek-out-of-nSystems,IEEETransactionsonReliability,58,1,pp.88–93.Barlow,R.E.andProchan,F.(1975).StatisticalTheoryofReliabilityandLifeTesting.ProbabilityModels(HoltRinehartandWinston).Barlow,R.E.andWu,A.S.(1978).Coherentsystemswithmulti-statecompo-nents,MathematicsofOperationsResearch,3,4,pp.275–281.Chang,G.J.,Cui,L.andHwang,F.K.(2000).ReliabilitiesofConsecutive-kSystems,NetworkTheoryandApplications,Volume4(KluwerAcademicPublishers). 152September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book138ReliabilityModelingwithApplicationsEl-Neweihi,E.,ProschanF.andSethuraman,J.(1978).Multi-statecoherentsystem,JournalofAppliedProbability,15,pp.675–688.Griffith,W.S.(1980).Multistatereliabilitymodels,JournalofAppliedProbabil-ity,17,pp.735–744.Haim,M.andPorat,Z.(1991).Bayesreliabilitymodelingofamultistateconsecu-tivek-out-of-n:Fsystem,ProceedingAnnualReliabilityandMaintainabilitySymposium,pp.582–586.Huang,J.,Zuo,M.J.andWu,Y.H.(2000).Generalizedmulti-statek-out-of-n:Gsystems,IEEETransactionsonReliability,49,1,pp.105–111.Kolowrocki,K.(2004).ReliabilityofLargeSystems(Elsevier).Kossow,A.andPreuss,W.(1995).Reliabilityoflinearconsecutively-connectedsystemswithmultistateComponents,IEEETransactionsonReliability,44,3,pp.518–522.Koutras,M.V.(1997).Consecutive-k,r-out-of-n:DFMsystems,MicroelectronicsandReliability,37,4,pp.597–603.Kuo,W.andZuo,M.J.(2003).OptimalReliabilityModeling,PrinciplesandApplications(JohnWileyandSons).Malinowski,J.andPreuss,W.(1995).Reliabilityofcircularconsecutively-connectedsystemswithmulti-statecomponents,IEEETransactionsonReliability,44,3,pp.532–534.Malinowski,J.andPreuss,W.(1996).Reliabilityofreverse-Tree-Structuredsys-temswithmulti-statecomponents,MicroelectronicsandReliability,36,1,pp.1–7.Rushdi,A.M.(1986).Utilizationofsymmetricswitchingfunctionsinthecompu-tationofk-out-of-nsystemreliability,MicroelectronicsandReliability,26,5,pp.973–987.Tian,Z.,Zuo,M.J.andYam,R.CM.(2009).Multi-statek-out-of-nsystemsandtheirperformanceevaluation,IIETransactions,41,pp.32–44.Yamamoto,H.,Akiba,T.andNagatsuka,H.(2006).EfficientmethodsforthesystemstatedistributionofMulti-statek-out-of-n:GSystems,TheJournalofReliabilityEngineeringAssociationofJapan,28,5,pp.395–404.(inJapanese)Yamamoto,H.,Akiba,T.andNagatsuka,H.(2009).CalculatingmethodforthesystemstatedistributionofMulti-statek-out-of-n:FSystems,IEICETransactionsonFundamentalsofElectronics,CommunicationsandCom-puterSciences,E92-A,7,pp.1593–1599.Yamamoto,H.,Akiba,T.,Yamaguchi,T.andNagatsuka,H.(2011).Anevaluat-ingalgorithmforthesystemstatedistributionsofgeneralizedmulti-statek-out-of-n:FSystems,JournalofJapanIndustrialManagementAssocia-tion,61,6E,pp.347–354.Zuo,M.J.andLiang,M.(1994).Reliabilityofmultistateconsecutively-connectedsystems,ReliabilityEngineeringandSystemSafety,44,pp.173–176.Zuo,M.J.,Huang,J.andKuo,W.(2003).Chapter1:Multi-statek-out-of-nSystems,HandbookofReliabilityEngineering(Springer),pp.3–17.Zuo,M.J.andTian,Z.(2006).Performanceevaluationofgeneralizedmulti-statek-out-of-nSystems,IEEETransactionsonReliability,55,2,pp.319–327. 153September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookChapter8Multi-stateComponentsAssignmentProblemwithOptimalNetworkReliabilitySubjecttoAssignmentBudgetYi-KueiLinandCheng-TaYehDepartmentofIndustrialManagement,NationalTaiwanUniversityofScience&Technology,No.43,Sec.4,KeelungRd.,Da’anDist.,TaipeiCity106,Taiwan1IntroductionTheassignmentproblem(AP)isacriticalissueindecisionmaking.Atypi-calAPistoassignasetofcomponentstoasetoflocationsinasystemwithmaximaltotalprofitorminimaltotalcost.Forinstance,asetoftransmis-sionlines,suchascoaxialcables,fiberoptics,etc.,isreadytobeassignedtothegivenlocationsofaninformationnetwork.TheAPhasmanyexten-sionsinpracticalapplications:facilitylocation,taskassignment,personnelscheduling,andsoon[Winston(1995)].[Pentico(2007)]surveyedmanytypicalAPsandmadethecategorizationforthem,buttheseproblemsfo-cusedonmaximizingthetotalprofitorminimizingthetotalcostwithouttakingthefailureofthecomponentintoconsideration.Inpractice,eachcomponentshouldbemultistateduetofailure,maintenanceandpartiallyfailure.Thatis,eachcomponentownsseveralpossiblecapacitieswithaprobabilitydistributionandmayfail.Asthesetofmultistatecomponentsisassignedtothearcs(i.e.locations)ofasystem,eacharcismultistate.Suchasystemisthereforetreatedasastochastic-flownetwork(SFN)withasetofarcsandnodes.Thenthenetworkreliabilityisdefinedtobetheprobabilitythatthedunitsofhomogenouscommodityaretransmitted139 154September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book140ReliabilityModelingwithApplicationssuccessfullyfromthesourcenodestothesinknodet.Manystudieseval-uatedthenetworkreliabilityintermsofminimalpaths(MPs)[Lin,JaneandYuan(1995)].AnMPisanordersequenceofarcsfromstotthathasnocycle.Inadditiontoevaluatingthenetworkreliability,severalissuesaboutthenetworkreliabilityoptimizationhavebeenexploredrecently[HsiehandChen(2005);LevitinandLisnianski(2001);LisnianskiandLevitin(2003);Liu,Zhang,MaandZhao(2007);PaintonandCampbell(1995)].Accord-ingly,[HsiehandChen(2005)]developedanupdatingschematodeterminetheoptimalmultistatemulti-terminalnetworkreliabilityunderarangeofdemandunderthedemand-dependentanddemand-independentcostcon-straints.[Liuetal.(2007)]adoptedgeneticalgorithm(GA)toevaluatetheoptimalmultistatemulti-terminalnetworkreliabilityunderflowas-signmentsofmulti-commodity.[LisnianskiandLevitin(2003)]classifiedthenetworkreliabilityoptimizationproblemsintotwocategories:achiev-ingtheoptimalnetworkreliabilityunderdifferentconstraints[PaintonandCampbell(1995)]andminimizingtheresourcesrequiredsubjecttoanet-workreliabilitylevel[LevitinandLisnianski(2001)].Theprecedingliteraturesmajorinthetopicsofcommodityallocation,flowassignment,ornetworkstructure.Howtoassignthemultistatecom-ponentssuchthatthenetworkreliabilityismaximalisneverdiscussed.Thus,thisarticlestudiesthemultistatecomponentsassignmentproblem(MCAP)withoptimalnetworkreliabilitysubjecttotheassignmentbud-getfordesirableandpracticalrequirements.Inthisproblem,thenetworktopologyisimmobileandthereisasetofcomponentsabletobeassignedtothenetwork’sarcs.Eachcomponenthasanassignmentcost,thecostofassigning/purchasingthecomponent.Intuitively,theimplicitenumera-tionmethod(IEM)maysearchesfortheoptimalcomponentsassignmentwithmaximalnetworkreliability.However,thismethodisinefficientwhenthenetworkislarge.GAisapowerfulprobabilisticsearchandoptimiza-tionalgorithm.Manyresearchers[ChuandBeasley(1997);Harper,DeSenna,VieiraandShahani(2005);MajumdarandBhunia(2007);Wang(2002);Wilson(1997)]haveemployedGAtosolvesomeversionsofAPs.Moreover,GAisappliedtosolvenotonlyAPs,butalsothenetworkreli-abilityoptimizationproblems[LevitinandLisnianski(2001);Liu,Zhang,MaandZhao(2007);PaintonandCampbell(1995)].Therefore,thisarti-clereferstothestandardGAfrom[Goldberg(1989)]andproposesaGAbasedalgorithmtodeterminetheoptimalcomponentsassignmentwith 155September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookMulti-stateComponentsAssignmentProblem141maximalnetworkreliabilitysubjecttotheassignmentbudget.IntheGA-basedalgorithm,acomponentsassignmentisrepresentedasachromosome(solution),andthefitnessfunctionistoevaluatethenetworkreliabilityofachromosomeintermsofMPsandstate-spacedecomposition.TheexperimentalresultsshowthattheGA-basedalgorithmisexecutedinareasonabletime.2AssumptionsLet(N,A)beanSFNwithasinglesourcesandasinglesinkt,whereNdenotesthesetofnodesandA={ai|1≤i≤n}denotesthesetofndirectedarcsconnectingnodes.SupposetherearetotallymMPs,Λ1,Λ2,···,Λm.LetΩ={ωk|1≤k≤z}denotethesetofcomponents.Eachcomponentωkhasmultiplestates,1,2,···,Mk,withcorrespondingcapacities,0=hk(1) 156September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book142ReliabilityModelingwithApplicationsunderthecomponentsassignmentBifandonlyiffj≤min{hbi(Mbi)},j=1,2,···,m,(2)ai∈Λj∑fj≤{hbi(Mbi)},i=1,2,···,n,(3)ai∈Λjwhereminai∈Λj{hbi(Mbi)}isthemaximalcapacityofΛj.Constraint(2)representthattheflowthroughΛj,andconstraint(3)meansthesumofflowthrougharcaicannotexceedthemaximalcapacityofcomponentωkassignedtoarcai.Similarly,anyFisfeasibleunderXifandonlyiffj≤min{xi},j=1,2,···,m,(4)ai∈Λj∑fj≤xi,i=1,2,···,n,(5)ai∈Λj∑mInotherwords,themaximalflowunderXisV(X)≡max{j=1fj|FisfeasibleunderX}.3.1NetworkReliabilityEvaluationLetTBbethesetofstatevectorsunderthecomponentsassignmentB.ThenetworkreliabilityunderBdenotedbySRd(B)isdefinedastheprobabilitythatdunitsofhomogenouscommoditycanbesuccessivelydeliveredfromstotunderB,i.e.,SRd(B)≡Pr{V(X)≥d,X∈TB}.LetXB={X|V(X)≥d,X∈TB},thenSRd(B)canbeobtainedbysumminguptheprobabilitiesofallX∈XB.Thus,∑SRd(B)=Pr(X).(6)X∈XBGenerally,itisnotawisewaytoenumerateallX∈XBandthentosumuptheirprobabilitiesinordertoobtainSRd(B).Instead,thisarticleevaluatesSRd(B)basedonthemethodproposedby[Linetal.(1995)].AnyminimalstatevectorinthesetXBiscalledalowerboundarypointfordord-MP.Ad-MPXmeansthatV(X)≥dandV(X) 157September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookMulti-stateComponentsAssignmentProblem143Suchaprobabilitycanbecalculatedbyinclusion-exclusionprinciple[Lin(2001)]orstate-spacedecomposition[Jane,LinandYuan(1993)].JaneandLaih(2008)validatedthemodifiedversionhasbetterefficiencyincomputa-tionandstoragespace.Thus,thisarticleadoptsJaneandLaih’sstate-spacedecompositiontoevaluatethenetworkreliability.3.2Generatealld-MPsTheflowvectorFmeetstheexactdemandifitsatisfiesconstraints(2),(3)and(8),∑mfj=d,(8)j=1whereconstraint(8)showsthatthesumofflowfromstotneedstomeetdemandd.LetF={F|Fsatisfiesconstaints(2),(3)and(8)}.IfX∈TBisad-MP,thenthereexistsanfeasibleF∈FunderXsuchthatxi=hbi(l)whereal∈{1,2,···,Mbi}suchthat∑hbi(l−1) 158September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book144ReliabilityModelingwithApplications∪Step4.IfXj 159September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookMulti-stateComponentsAssignmentProblem145bedefinedandthechromosomeshouldbeencoded.Thepopulationsizerepresentsthenumberofchromosomesinthepopulation.ThenumberofgenerationistheterminalconditionoftheGA-basedalgorithm.Thecrossoverratecontrolstheprobabilityofexecutingcrossover.Themutationratecontrolstheprobabilityofexecutingmutation.Thisarticleutilizestheintegerencodingtorepresentachromosomesothatthechromosomefitsinwiththecomponentsassignment(i.e.chromo-someGisequivalenttocomponentsassignmentB).Hence,achromosomeisdenotedasG=(g1,g2,···,gn),wheregi∈{1,2,···,z}signifiesthatarcaiisgivenwithcomponentωkifgi=k.Suchanencodingsatisfiesconstraints(11)and(12),andhaslessmemoryutilizationthanthebinaryencodingwhenthenetworkislarge.TheGA-basedalgorithmstartswithapopulation,inwhichthereare(Psize)chromosomes.Subsequently,allchromosomesmustbeevaluatedbythefitnessfunction.Tobeworthyofattention,thefitnessvalueofachromosomeisequaltothenetworkreliability.However,ifthetotalas-signmentcostofachromosomeexceedtheassignmentbudgetorthereisnoX∈XG,thenthenetworkreliabilityofthechromosomeisgivenwitharandompenaltyvaluethatissuggestednottoexceed10−4.Thenetworkreliabilityofeachchromosomeiscalculatedbythefollowingsteps.AlgorithmII.Step1.Determinewhetherthechromosomesatisfiestheconstraint(14)ornot,∑ncgi≤C.(14)i=1Ifthechromosomedoesnotsatisfytheconstraint(14),givearan-dompenaltyvaluetobeitsnetworkreliability,andthenevaluatethenextchromosome.Step2.FindallFsatisfyingthefollowingconstraints.fj≤min{hgi(Mgi)},j=1,2,···,m,(15)ai∈Λj∑fj≤{hgi(Mgi)}i=1,2,···,n,and(16)ai∈Λj∑mfj=d.(17)j=1 160September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book146ReliabilityModelingwithApplicationsIfnofeasibleFexists,givearandompenaltyvaluetobeitsnetworkreliability,andthenevaluatethenextchromosome.Step3.TransformeachFintoXviathefollowingequation.xi=hgi(l)wherel∈{1,2,···,Mgi}suchthat∑hgi(l−1) 161September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookMulti-stateComponentsAssignmentProblem147geneexchangeswiththe4thgene.Thus,thechromosomebecomes(231456).Ifthevalueofthe2ndgenechangesinto7,thenthechromosomeis(271356)duetonoduplicategeneinthechromosome.Throughthemodifiedsingle-pointcrossoverandthemodifieduniformmutation,theduplicategenescanbeavoided,andtheassumptionthateachcomponentcanbeassignedtoatmostonearcandeacharcmustbegivenwithexactonecomponentcanbesatisfied.AlgorithmIII.Step1.DeterminePsize,Pcr,Pmr,andgtime,andgenerateaninitialran-dompopulation.Step2.EvaluatenetworkreliabilityforeachchromosomeinpopulationusingAlgorithmII.Step3.Iftheterminalconditionissatisfied,returntheoptimalsolutioninthecurrentgeneration,elsegotostep4.Step4.ExecutetheevolutionprocesstoproducePsizechromosomes.4.1Utilizeroulettewheelselection.4.2.Implementthemodifiedsingle-pointcrossoverbasedonPcr.4.3.ExecutethemodifieduniformmutationbasedonPmr.Step5.EvaluatenetworkreliabilityforthePsizechromosomesfromstep4usingAlgorithmII.Step6.MixtheoriginalPsizechromosomesandthenewPsizechromo-somes.Subsequently,choosetheamountofbetterPsizechromo-somesfromthe2×Psizechromosomestobethenewpopulationaccordingtothenetworkreliability,andthengotostep3.6NumericalExperimentsTwoillustrativeexamplesareadoptedtodemonstratetheGA-basedalgo-rithm.TheGA-basedalgorithmandtheIEMareprogrammedinMATLABandimplementedonapersonalcomputerwithIntelCore2QuadCPU3.0Gand2GRAM.6.1Example1Inexample1,theGA-basedalgorithmiscomparedwiththeIEMbasedonthenetworkreliabilityandCPUtimethroughasimplenetwork(aspresentedinFig.1).AllMPsofthenetworkareΛ1={a1,a2},Λ2= 162September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book148ReliabilityModelingwithApplicationsFig.1.1Asimplenetwork.Fig.1Asimplenetwork.Table1Probabilitydistributionofcapacitiesfor10components.CapacityComponents(ωk)Cost05101520251600.050.10.250.60a02100.10.30.60003200.10.900004500.10.900005600.10.900006200.050.250.70007500.3000.7008800.150.050.20.10.5091000.30.050.0500.050.5510700.050.250.200.10.4aThecomponentdoesnotprovidethiscapacity.{a1,a3,a6},Λ3={a2,a4,a5}andΛ4={a5,a6},andthereare10compo-nentsreadytobeassigned.Eachcomponentprovidesvariouscapacitieswithaprobabilitydistribution(seeTable1).Thepossiblecapacityofeachcomponentisnotformedwithaconsecutiverangeofintegers.Severalex-perimentsareimplementedbytheGA-basedalgorithmwithPsize=30,Pcr=0.9,Pmr=0.05andgtime=15todiscusstheresultsofvariousdemandsandbudgets.Fromtheexperimentalresults(seeTable2),theGA-basedalgorithmnotonlyobtainstheoptimalsolutions,butalsohasbettercomputationalefficiencythantheIEM.Thebudgethasobviousinflu-enceonthenetworkreliability.Underthesamedemandlevel,thenetworkreliabilityusuallydecreaseswhenthebudgetdecreases,especiallyunder 163September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookMulti-stateComponentsAssignmentProblem149Table2Probabilitydistributionofcapacitiesfor10components.IEMAlgorithmIIIaMaximalExactsolution/CPUOptimalCPU(d,C)reliability♯optimaltimebsolutiontimenetworksolutions(5,240)0.991848(6,1,2,5,10,3)/21618.45(10,3,2,5,1,6)0.17(10,240)0.935256(1,10,5,3,2,6)/825.14(1,6,3,5,2,10)0.27(15,240)0.761084(1,8,2,3,6,7)/218.94(8,1,3,2,7,6)0.25(20,240)0.549362(1,10,2,3,6,7)/520.05(1,10,3,2,6,7)0.25(25,240)0.32487(1,6,2,3,8,7)/816.19(7,8,2,3,6,1)0.28(30,240)0.1575(1,10,3,2,7,6)/519.13(1,6,2,3,7,10)0.27(35,240)0∅c18.19∅0.02(5,210)0.988515(6,2,4,7,3,1)/2414.16(1,3,2,7,4,6)0.13(10,210)0.8947(2,7,4,3,6,1)/819.91(6,1,3,4,2,7)0.17(15,210)0.637245(2,7,4,3,6,1)/813.98(6,1,4,3,2,7)0.16(20,210)0.465675(7,1,4,3,2,6)/618.25(7,1,4,3,2,6)0.11(25,210)0.1764(2,6,4,3,1,7)/814.83(7,1,4,3,6,2)0.16(30,210)0∅17.03∅0.03(5,200)0∅10.24∅0.02aAlgorithmIIIparameters:Psize=30,Pcr=30,Pmr=0.05,gtime=15.bUnit:second.cNosolution.thegreatdemandlevel.Inthiscase,thereisnooptimalsolutionunderanydemandlevelifbudgetCislessthan210.6.2Example2Alargelogisticsserviceprovider(LSP)deploysitslogisticsnetworkasFig.2inwhichtherearesixMPs.TheLSPwilldeliver600unitsofhomogenouscommodityfromstot.However,theclientcanpayatmost2600dollars.Hence,theLSPneedstoplantheoptimalutilizationofitstransportationresourcestoobtainthemaximalnetworkreliabilityundertheexpensewhichtheclientcanpay.Tables3-5presentsthetransportationresourcestheLSPowns.Eachtransportationresourcehasanassignmentcost.Thecapacityofeachtransportationresourcerepresentsload-carryingabilityandmayhavemultiplestates.TheLSPutilizesAlgorithmIIIwithPsize=100,gtime=500,Pcr=0.6andPmr=0.1todeterminethebestsolutionaccordingtotheLSP’strans-portationresourcesandtheclient’srequirement.TheLSPimplements 164September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book150ReliabilityModelingwithApplicationsFig.2Alogisticsnetwork.AlgorithmIIIfor30timesandwantstoobtainthelargestandaveragemaximalnetworkreliabilityinthe30experiments.Thelargestmaximalnetworkreliabilityis0.936308withthecost2600andtheaveragemaximalnetworkreliabilityis0.92718.Moreoverinthese30experiments,theav-erageCPUtimeis761seconds.Obviously,AlgorithmIIIobtainsthebestsolutioninareasonabletime.Theprogressofsearchingfortheapproxi-matelyoptimalsolutionwiththemaximalnetworkreliability0.936308isdisplayedinFig.3.Theoptimalassignmentis(88,71,21,55,6,41,91,5,51,1,94,44,84,34,13,92,42,82,12,38,62,29,93,3,61,11,85,63,32,60,10,35,79). 165September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookMulti-stateComponentsAssignmentProblem151Table3Probabilitydistributionofcapacitiesfor100components.CapacityωkCost02004006008001000120011500.0100.0100.0100.972800.050.050.050.150.20.503650.070.0800.850004300.700000.3051100.01000.05000.9461800.01000.01000.987400.50.5000008350.250.250.500009550.150.250.10.10.10.10.2107000.050.050.900011250.010.990000012900.0200.0500.0500.8813500.0700.28000.65014950.050.050.9000015500.60.40000016800.150000.850017650.10.10.10.700018700.700000.3019500.070.180.75000020150.40.40.20000211200.010.0100.0300.950221600.02000000.98231000.050.05000.10.10.724350.70.30000025400.300.7000026800.10.10.8000027100.80.2000002850.90.1000002911500.0500.0500.10.830550.030.270.30.4000311300.010.010.020.020.020.020.932600.05000000.9533300.10.90000034200.080.370.55000035750.020.030.050.9000 166September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book152ReliabilityModelingwithApplicationsTable4(cont.ofTable3)Probabilitydistributionofcapacitiesfor100components.CapacityωkCost020040060080010001200361400.05000.05000.937800.070.2300.2000.538350.02000.0800.9039900.50.500000401000.7000.300041800.010.020.030.040.050.060.7942400.070.070.070.070.070.65043150.150.150.150.150.40044130000000.10.945600.020.480.50000461550.010.020.030.040.0500.8547950.050.30.65000048900.4000.2000.449850.5000.500050500.10.20.70000511500.0100.0100.0100.9752800.050.050.050.150.20.5053650.070.0800.8500054300.700000.30551100.01000.05000.94561800.01000.01000.9857400.50.50000058350.250.250.5000059550.150.250.10.10.10.10.2607000.050.050.900061250.010.990000062900.0200.0500.0500.8863500.0700.28000.65064950.050.050.9000065500.60.40000066800.150000.850067650.10.10.10.700068700.700000.3069500.070.180.75000070150.40.40.20000 167September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookMulti-stateComponentsAssignmentProblem153Table5(cont.ofTable4)Probabilitydistributionofcapacitiesfor100components.CapacityωkCost020040060080010001200711200.010.0100.0300.950721600.02000000.98731000.050.05000.10.10.77450.70.30000075400.300.7000076800.10.10.8000077100.80.2000007850.90.1000007911500.0500.0500.10.880550.030.270.30.4000811300.010.010.020.020.020.020.982600.05000000.9583300.10.90000084200.080.370.55000085750.020.030.050.9000861400.05000.05000.987800.070.2300.2000.588350.02000.0800.9089900.50.500000901000.7000.300091800.010.020.030.040.050.060.7992400.070.070.070.070.070.65093150.150.150.150.150.40094130000000.10.995600.020.480.50000961550.010.020.030.040.0500.8597950.050.30.65000098900.4000.2000.499850.5000.5000100500.10.20.700007ConclusionsTheMCAPisdifferentfromthetypicalAPsthatfocusonmaximizingthetotalprofitorminimizingthetotalcost.AnefficientalgorithmisdevelopedbasedonGA,inwhichthefitnessfunctionevaluatesthenetworkreliabilityofacomponentsassignmentintermsofMPsandstate-space 168September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book154ReliabilityModelingwithApplicationsFig.1.3Theprogressofsearchingfortheoptimalsolution.Fig.3Theprogressofsearchingfortheoptimalsolution.decomposition.Forthelargernetworkwith33arcsand100components,theexperimentalresultsshowthattheapproximatelyoptimalsolutioncanbefoundinnomorethan1544.Hence,AlgorithmIIIcanbeexecutedinareasonabletime.Throughtheobservationfromexample1,theremaybeseveraloptimalsolutionswithdifferenttotalassignmentcostsunderdemanddandbud-getC.Generally,thedecisionmakerfocusesondeterminingtheoptimalsolutionwithmaximalnetworkreliabilityandminimalassignmentcost.Therefore,theaddressedproblemcanbeextendedtothemulti-objectiveoptimizationproblem.ReferencesChu,P.C.andBeasley,J.E.(1997).Ageneticalgorithmforthegeneralizedassignmentproblem,ComputersandOperationsResearch24,pp.17–23.Ford,L.R.andFulkerson,D.R.(1962).Flowsinnetworks(PrincetonUniversity,NewJersey).Goldberg,D.(1989).GeneticAlgorithmsinSearch,OptimizationandMachineLearning,Reading(Addison-Wesley,Massachusetts).Harper,P.R.,DeSenna,V.,VieiraI.T.andShahani,A.K.(2005).Ageneticalgorithmfortheprojectassignmentproblem,ComputersandOperationsResearch32,pp.1255–1265. 169September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookMulti-stateComponentsAssignmentProblem155Hsieh,C.C.andChen,Y.T.(2005).Resourceallocationdecisionsundervariousdemandsandcostrequirementsinanunreliableflownetwork,ComputerandOperationsResearch32,pp.2771–2784.Jane,C.C.andLaih,Y.W.(2008).Apracticalalgorithmforcomputingmulti-statetwo-terminalreliability,IEEETransactionsonReliability57,pp.295–302.Jane,C.C.,Lin,J.S.andYuan,J.(1993).Reliabilityevaluationofalimited-flownetworkintermsofminimalcutsets,IEEETransactionsonReliability42,pp.354–361.Levitin,G.andLisnianski,A.(2001).Anewapproachtosolvingproblemsofmulti-statesystemreliabilityoptimization,QualityReliabilityEngineeringInternational17,pp.93–104.Lin,Y.K.(2001).Asimplealgorithmforreliabilityevaluationofastochastic-flownetworkwithnodefailure,ComputerandOperationsResearch28,pp.1277–1285.Lin,J.S.,Jane,C.C.andYuan,J.(1995).Onreliabilityevaluationofacapacitated-flownetworkintermsofminimalpathsets,Network25,pp.131–138.Lisnianski,A.andLevitin,G.(2003).Multi-statesystemreliability,assessment,optimizationandapplication,Vol.6(WorldScientific,Singapore).Liu,Q.,Zhang,H.,Ma,X.andZhao,Q.(2007).Geneticalgorithm-basedstudyonflowallocationinamulticommoditystochastic-flownetworkwithunreli-ablenodes,inProc.theEighthACISInternationalConferenceonSoftwareEngineering,ArtificialIntelligence,Networking,andParallel/DistributedComputing,pp.576–581.Majumdar,J.andBhunia,A.K.(2007).Elitistgeneticalgorithmforassignmentproblemwithimprecisegoal,EuropeanJournalofOperationalResearch177,pp.684–692.Painton,L.andCampbell,J.(1995).Geneticalgorithmsinoptimizationofsystemreliability,IEEETransactionsonReliability44,pp.172–178.Pentico,D.W.(2007).Assignmentproblems,agoldenanniversarysurvey,Euro-peanJournalofOperationalResearch176,pp.774–793.Wang,Y.Z.(2002).Anapplicationofgeneticalgorithmmethodsforteacherassignmentproblems,ExpertSystemswithApplications22,pp.295–302.Wilson,J.M.(1997).Ageneticalgorithmforthegeneralizedassignmentproblem,JournaloftheOperationalResearchSociety48,pp.804–809.Winston,W.L.(1995).IntroductiontoMathematicalProgramming:ApplicationandAlgorithms(Duxbury,California). 170May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank 171September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookChapter9ReliabilityAnalysisofaServerSystemwithReplicationSchemesMitsutakaKimuraDepartmentofInternationalCultureStudies,GifuCityWoman’sCollege,7-1Hitoichibakita-machi,Gifu501-0192,Japan1IntroductionRecently,theserversystemwithabackupsitehasbeenwidelyusedtoprotectenterprisedatabase.ThebackupsitestandsbythealertwhenamainsitehasbrokendownduetoDoSattack,electricityfailure,hurricanes,earthquakes,andsoon.Whenamainsitehasbrokendown,theserversys-temmigratestheroutineworkfromthemainsitetoabackupsite.Theserverinthemainsitetransmitsthedatabasecontentfromthemainsitetothebackupsiteusinganetworklink.Thisiscalledreplication[Yam-ato,KanandKikuchi(2006);Imai,Araki,Sugiura,FujitaandSuemura(2004);Nakamura,Fujiyama,KawaiandSunahara(2007);VERITASSoft-warecorporation].Thischaptersummarizesreliabilityanalysisofvariousreplicationschemes.InSection2,weconsideraserversystemwithreplicationusingjour-nalingfiles.Theservertakescheckpointforalldatabasetransactionsinthemainsiteandtransmitsthedatabasecontentfromthemainsitetothebackupsite.Thereplicationshouldexecutewheneverthestoragedatabaseinthemainsiteisupdated.Butthereplicationgenerallyexecutesatreg-ularintervalsbecauseithasaprohibitivecost.However,ithasaproblemtocompromiseconsistencyofdatabasecontent.Therefore,theserverinthemainsitetransmitsjournalingfiles(updatefilesandupdatehistory)tothebackupsiteassoonastheserverupdatesthestoragedatabasewhen157 172September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book158ReliabilityModelingwithApplicationsaclientrequeststhedataupdate.Whentheserverinthebackupsitereceivesjournalingfiles,theyarenotimmediatelyreflectedinthedatabasetopreventdelay.Whentheserverinthemainsitebreaksdown,theserversystemmigratestheroutineworkfromthemainsitetoabackupsiteandthestateofadatabaseisrolledbacktothemostrecentcheckpointandrestoreaconsistentstatebyreadingbackjournalingfiles[Yamato,KanandKikuchi(2006);FujitaandYata(2005);Watabe,Suzuki,MizunoandFujiwara(2007);GodaandKitsuregawa(2007)].Whenthedatabaserestoresaconsistentstatebyusingsomejournalingfiles,themorejournalingfilesthedatabasehas,ittakestoomuchtimetoreadbackthem,andtheserversystemsufferslossescausedbylowerefficiencywhileitwasdown.Theyhavebeenpointedoutasaproblemtobestudied.Inreference[FujitaandYata(2005)]and[Watabe,Suzuki,MizunoandFujiwara(2007)],theyhaveproposedthetechniquetospeedupthedatabaseprocessingtorestoreaconsistentstatebybeingreadbackjournalingfiles.Inreference[GodaandKitsuregawa(2007)],theyhaveproposedthemethodtoreduceoperationalcostinadisasterrecoverysys-tembasedonlogforwardingremotecopy.Weformulateastochasticmodelofaserversystemwithasynchronousreplicationusingjournalingfilesandproposedanoptimalpolicytoreducewasteofcostsforreplicationandjournalingfiles[Kimura,ImaizumiandNakagawa(2010,2011)].InSection3,weconsiderconsideraserversystemwithreplicationbufferingrelaymethod.Theserversystemconsistsofabufferingrelayunitaswellasbothmainandbackupsites.Theserverinamainsiteupdatesthestoragedatabasewhenaclientrequeststhedataupdate,andtransfersthedataandtheaddressofthephysicallocationupdateddataonthestoragetoabufferingrelayunit.Theservertransmitsallofthedatainthebufferingrelayunittoabackupsiteatanytime(replication).Whenamainsitehasbrokendown,theroutineworkismigratedfromthemainsitetothebackupsiteandisabletobeexecutedimmediately.Whenaclientrequeststhedataupdateorthedateread,theserverconfirmtheaddresstableheldinthebufferingrelayunit.Iftherequesteddatacorrespondswiththedataheldinthebufferingrelayunit,itistransmittedfromthebufferingrelayunittothebackupsiteinstantly.Then,thewasteofcostfortransmittingtheupdateddatahasbeenpointedout[Kan,Yamato,KanekoandKikuchi(2005)].Weconsidertheproblemofreliabilityinaserversys-temusingthereplicationbufferingrelaymethodinordertoreducethecostofreplicationandtransmittingtheupdateddatainbufferingrelayunit.Thatis,weformulatethestochasticmodelofaserversystemwith 173September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisofaServerSystemwithReplicationSchemes159thereplicationbufferingrelaymethodanddiscussanoptimalreplicationintervaltominimizeit[Kimura,ImaizumiandNakagawa(2011,2012)].2ServerSystemwithReplicationusingJournalingFilesThissectionformulatesastochasticmodelofaserversystemwithreplica-tionconsideringthenumberoftransmittingjournalingfiles.Thatis,theserverupdatesthestoragedatabaseandtransmitsjournalingfileswhenaclientrequeststhedataupdate.Theservertransmitsthedatabasecontenttoabackupsiteeitherataconstanttimeorafteraconstantnumberoftransmittingjournalingfiles.Weproposeanoptimalpolicytoreducewasteofthecostforexecutionoftransmittingjournalingfiles.Wederivetheex-pectednumbersofthereplicationandofjournalingfiles.Furthermore,wecalculatetheexpectedcostanddiscussanoptimalreplicationintervaltominimizeit.Finally,numericalexamplesaregiven.2.1ReliabilityQuantitiesAserversystemconsistsofamonitor,amainsiteandabackupsiteasshowninFig.1.Bothmainandbackupsitesconsistofidenticalserverandstorage.Thebackupsitestandsbythealertwhenthemainsitebreaksdown.Theserverinthemainsiteperformstheroutineworkandupdatesthestoragedatabasewhenaclientrequeststhedataupdate.Then,themonitortransmitstheupdatefilesandupdatehistory(journalingfiles)fromthemainsitetotheMonitorMainsiteBackupsiteServerServerStorageStorageClientFig.1Outlineofaserversystem. 174September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book160ReliabilityModelingwithApplicationsMainSiteOccurrenceofdisasterDataUpdateServerReplicationReplicationStorageUpdateFilesandUpdateHistoryBackupSite(Journalfiles)RollbackServerReadingbackjournalfilesStorageTimeTk=3CheckpointFig.2Operationofanasynchronousreplicationschemeusingjournalingfiles.backupsiteimmediately.Furthermore,themonitororderstheservertotransmitthedatabasecontenttothebackupsiteeitherataconstanttimeTorafteraconstantnumberk(k=1,2,···)oftransmittingjournalingfiles.TheoperationofaserversystemwithasynchronousreplicationschemeusingjournalingfilesisshowninFig.2.Weformulatethestochasticmodelasfollows:(1)Whentheserverinthemainsitebreaksdownbyfailuressuchasadisasterandcrashfailureandsoon,theroutineworkismigratedfromthemainsitetothebackupsite,andthestateofadatabaseisrolledbacktothemostrecentcheckpointandcanrestoreaconsistentstatebyreadingbackjournalingfiles(systemmigration).Bothmonitorandbackupsitedonotbreaksdown,however,theserverinthemainsitebreaksdownaccordingtoageneraldistributionF(t).(2)Aclientrequeststhedataupdatetothestorage.TherequestrequiresthetimeaccordingtoageneraldistributionA(t),andthetotalofup-dateandtransmissionofjournalingfilesrequirethetimeaccordingtoageneraldistributionB(t).(3)Themonitororderstheservertotransmitsthedatabasecontentfromthemainsitetothebackupsite,aftertheserverinthemainsiteupdatesthedatabyrequestingfromclientandtransmitsjournalingfilesateitherk(k=1,2,···)timesortimeT(0≤T<∞),whereG(t)≡0fort 175September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisofaServerSystemwithReplicationSchemes161sitetothebackupsite(replication).ThereplicationrequiresthetimeaccordingtoageneraldistributionW(t).(b)Iftheserverinthemainsitebreaksdownwhiletheservertransmitsthedatabasecontentfromthemainsitetothebackupsite,themonitorimmediatelyorderstomigratetheroutineworkfromthemainsitetothebackupsiteandthesystemmigrationisexecuted.Undertheaboveassumptions,wedefinethefollowingstatesoftheserversystem:State0:Systembeginstooperateorrestart.State1:WhenthemonitorconfirmsthestateofthemainsiteattimeT,thereplicationprocessbeginstoexecute.State2:Whentheserverinthemainsitehastransmittedjournalingfilesatk(k=1,2,···)times,thereplicationprocessbeginstoexecute.State3:Whentheserverinthemainsitebreaksdown,theprocessofadatabaseisrolledbacktothemostrecentcheckpointandcanrestoreaconsistentstatebyreadingbackjournalingfiles.ThesystemstatesdefinedaboveformaMarkovrenewalprocessandrepresenttimepointsoftransition[Osaki(1992);Yasui,NakagawaandSandoh(2002)],wherestate3isanabsorbingstateandisafailurestate.AtransitiondiagrambetweensystemstatesisshowninFig.3.LetΦ∗betheLaplace-Stieltjes(LS)transformofanyfunctionΦ(t),∫∞i.e.,Φ∗≡e−stdΦ(t)forRe(s)>0.TheLStransformsoftransition0probabilitiesQi,j(t)(i=0,1,2;j=0,1,2,3)aregivenbythefollowing3102Fig.3Transitiondiagrambetweensystemstates. 176September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book162ReliabilityModelingwithApplicationsequations:Q∗(s)=e−sTF(T)[1−H(k)(T)],(1)0,1∫TQ∗(s)=e−stF(t)dH(k)(t),(2)0,20∫TQ∗(s)=e−st[1−H(k)(t)]dF(t),(3)0,30∫∞Q∗(s)=e−stF(t)dW(t)(i=1,2),(4)i,00∫∞Q∗(s)=e−stW(t)dF(t)(i=1,2),(5)i,30whereH(t)≡A(t)∗B(t),Φ(t)≡1−Φ(t)representsasurvivalfunctionofanyfunctionΦ(t),andΦ(i)(t)isthei-foldconvolutionofΦ(t)andΦ(i)(t)≡∫tΦ(i−1)(t)∗Φ(t)(i=1,2,···),Φ(t)∗Φ(t)≡Φ(t−u)dΦ(u),Φ(0)(t)≡112021fort≥0.WederivetheexpectednumberMDoftransmittingjournalingfilesuntilstate3.TheexpectednumberMD(t)in[0,t]isgivenbythefollowingrenewalequation:∑k∫tM(t)=(i−1)G(t)H(t)(i−1)∗H(t)dF(t)Di=10[∫]∑kt+(i−1)F(t)H(t)(i−1)∗H(t)dG(t)∗Q(t)1,3i=10[∫]t+kG(t)F(t)dH(t)(k)∗Q(t)2,30[∫]∑kt+F(t)H(t)(i−1)∗H(t)dG(t)∗Q(t)∗[1+M(t)]1,0Di=10[∫]t+G(t)F(t)dH(t)(k)∗Q(t)[1+M(t)].(6)2,0D0 177September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisofaServerSystemwithReplicationSchemes163Bysolving(6)forMD(t),itsLStransformis∑k∫Te−st[H(i)(t)−kH(k)(t)]dF(t)i=10{+e−sTF(T)∑k[H(i)(T)−kH(k)(T)]i=1}+k∫Te−stF(t)dH(k)(t)∫∞e−stW(t)dF(t)00M∗(s)={}.(7)D∫T1−e−sTF(T)[1−H(k)(T)]+e−stF(t)dH(k)(t)0∫∞×e−stF(t)dW(t)0Hence,theexpectednumberMDis∗Y(T,k)MD≡limMD(t)=limMD(s)=∫∞,(8)t→∞s→01−X(T,k)W(t)dF(t)0where∫TX(T,k)≡F(T)+H(k)(t)dF(t),0∑k∫TY(T,k)≡F(T)dH(i)(t)i=10[k∫]∫∑T∞−F(T)H(i)(T)+kH(k)(t)dF(t)W(t)dF(t).i=100Similarly,TheexpectednumberMR(t)in[0,t]isgivenbythefollowingrenewalequation:MR(t)=Q0,1(t)+Q0,2(t)+[Q0,1(t)∗Q1,0(t)+Q0,2(t)∗Q2,0(t)]∗MR(t).(9)TheLStransformM∗(s)oftheexpectednumberM(t)in[0,t]isRRQ∗(s)+Q∗(s)∗0,10,2MR(s)=1−Q∗(s)Q∗(s)−Q∗(s)Q∗(s).(10)0,11,00,22,0Hence,theexpectednumberMRis∗X(T,k)MR≡limMR(t)=limMR(s)=∫∞.(11)t→∞s→01−X(T,k)W(t)dF(t)0 178September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book164ReliabilityModelingwithApplications2.2OptimalPolicyWeproposeanoptimalpolicytoreducewasteofcostsforreplicationandjournalingfiles.Thatis,wecalculatetheexpectedcostuntilstate3andderiveoptimalreplicationintervalsT∗andk∗tominimizeit.LetcbeDthecostfortransmittingjournalingfilesandcRbethecostforreplication.WedefinetheexpectedcostC(T,k)asfollows:C(T,k)≡cDMD+cRMR.(12)WeseekanoptimalreplicationintervalT∗whichminimizesC(T,k)in(12)forcR≥cDandgivenk(k≥1).DifferentiatingC(T,k)withrespecttoTandsettingitequaltozero,∫∞∫∞cRQk(T)[1−X(T,k)W(t)dF(t)]−Y(T,k)W(t)dF(t)=,(13)00cDwhereY′(T,k)Qk(T)≡−,X′(T,k)andΦ′(t)≡dΦ(t)/dt.Denotingtheleft-handsideof(22)byL(T),k∫∞L′(T)=Q′(T)[1−X(T,k)W(t)dF(t)].(14)kk0′′Thus,ifQk(T)>0thenLk(T)>0because∫∞1−X(T,k)W(t)dF(t)>0.0Therefore,wehavethefollowingoptimalpolicy:(i)IfL(0)≥c/c,thenT∗=0.kRD(ii)IfLk(∞)>cR/cD>Lk(0),thenthereexistsafiniteanduniqueT∗(0 179September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisofaServerSystemwithReplicationSchemes165Clearly,α+wα+kwQk(0)=,Qk(∞)=λ+wλ+wSupposethatk=1.Then,Q(T)≡(α+w)/(λ+w),andhence,T∗=∞,ki.e.,weshoulddonoreplication.Next,fork≥2,k∑−1j−1∑ji′wλ(αT)(αT)2Qk(T)=λ+w∑k−1j2j!i!(j−i)>0,{[(αT)/j!]}j=0j=0i=0whichfollowsthatQk(T)increasesstrictlywithTfrom(α+w)/(λ+w)to(α+kw)/(λ+w).Thus,Lk(T)increasesstrictlywithTfromLk(0)=λ(α+w)/(λ+w)2toL(∞).kTherefore,wehavethefollowingoptimalpolicyfork≥2:(i)IfL(0)≥c/c,thenT∗=0andkRD()λ+wC(0,k)=cR.w(ii)IfL(∞)>c/c>L(0),then0 180September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book166ReliabilityModelingwithApplicationsDenotingtheleft-handsideof(16)byLT(k),[]Y(T,k+2)−Y(T,k+1)Y(T,k+1)−Y(T,k)LT(k)−LT(k−1)=−Z(k+2)−Z(k+1)Z(k+1)−Z(k)×Z(k+1).If−[Y(T,k+1)−Y(T,k)]/[X(T,k+1)−X(T,k)]increasesstrictlywithkandLT(∞)>(cR/cD)[(λ+w)/w],thenthereexistsafiniteanduniquek∗(1≤k∗<∞)whichsatisfies(16).Next,weconsidertheparticularcasethatA(t)=1−e−αt,B(t)=1,F(t)=1−e−λtandW(t)=1−e−wt.Then,Y(T,k+1)−Y(T,k)αwkλ−=+.X(T,k+1)−X(T,k)λλ+wλ+wTherefore,thereexistsafiniteanduniquek∗(1≤k∗<∞)whichsatisfies(16).Example9.1.WecomputenumericallyanoptimalreplicationintervalT∗whichmin-imizesC(T,k)in(12).Supposethatthemeantime1/βrequiredforthedataupdateisaunittime.Itisassumedthatthemeangenerationintervalofrequestthedataupdateis(1/α)/(1/β)=2∼10,themeangenera-tionintervalofaserverdownis(1/λ)/(1/β)=1000,10000,themeantimerequiredforthereplicationis(1/w)/(1/β)=100∼300,thenumberoftransmittingjournalingfilesisk=5.Further,weintroducethefollowingcosts:ThecostfortransmittingjournalingfilesiscD=1,thecostforreplicationiscR/cD=5,10.Table1givestheoptimalT∗whichminimizesthecostC(T,k).Forexample,whenk=5,cD=1,cR=5and1/β=1,1/α=10,1/w=200,1/λ=1000,theoptimalintervalisT∗=54.7.ThisindicatesthatT∗increasewithrespectto1/αand1/λ.Further,T∗increasewithrespecttocR/cD.Thus,weshouldtransmitmorejournalingfilesratherthanexecutereplication.Whenc/cislarge,T∗decreasewithrespectto1/λ.WhenRD1/λgetstoacertaininterval,weshouldexecutereplicationratherthantransmitjournalingfiles.Moreover,T∗decreasewithrespectto1/w.Inthiscase,weshouldexecutereplicationratherthantransmitjournalingfiles.Table2givestheoptimalreplicationintervalk∗whichminimizesthecostC(T,k).Forexample,whencD=1,cR=5and1/β=1,1/α=10,1/w=200,1/λ=1000,T=10.0,theoptimalreplicationintervalisk∗=5.Thisindicatesthatk∗increasewithrespectto1/λ.Further,k∗ 181September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisofaServerSystemwithReplicationSchemes167Table1OptimalreplicationtimeTtominimizeC(T,k)whenk=5β/λ100010000cR/cDβ/wβ/α2510251010010.927.857.211.128.157.7520010.526.754.711.027.756.230010.225.952.910.927.455.510011.229.563.311.329.662.91020010.727.858.911.128.559.230010.426.856.411.028.057.710013.643.1108.713.340.9101.65020012.236.590.212.234.882.230011.633.982.411.732.574.3Table2OptimalreplicationintervalktominimizeC(T,k)β/λcR/cDβ/wβ/α2(T=5.0)5(T=10.0)10(T=20.0)1001766520016553001655100176710001020017663001656100209750200198730018771001766520017663001756100186710000102001766300176610020975020019773001877increasewithcR/cD.Thus,weshouldtransmitmorejournalingfilesratherthanexecutereplication.Whenc/cislarge,k∗donotdependon1/λ.RDThisindicatesthatk∗decreasewithrespectto1/w.Inthiscase,weshouldexecutereplicationratherthantransmitjournalingfiles.Further,when1/λislarge,k∗donotdependon1/wandc/candbecomeconstant.RD 182September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book168ReliabilityModelingwithApplications3ServerSystemwithReplicationBufferingRelayMethodThissectionconsiderstheproblemofreliabilityinserversystemusingthemethodinordertoreducethewasteofcostsforreplicationandtransmittingtheupdateddatainbufferingrelayunit.Weformulateastochasticmodelofaserversystemwiththereplicationbufferingrelaymethodandderivetheexpectednumberofthereplicationandofupdateddatainbufferingrelayunit.Further,wecalculatetheexpectedcostanddiscussanoptimalreplicationintervaltominimizeit.Finally,anumericalexampleisgiven.3.1ReliabilityQuantitiesAserversystemconsistsofamonitor,abufferingrelayunit,amainsiteandabackupsiteasshowninFig.4.Bothmainandbackupsitesconsistofidenticalserverandstorage,andbothbackupsiteandbufferingrelayunitstandbythealertwhenadisasterhasoccurred.Theserverinthemainsiteperformstheroutineworkandupdatesthestoragedatabasewhenaclientrequeststhedataupdate.Then,themonitortransmitstheupdateddataandtheaddressofthephysicallo-cationupdateddataonthestoragetothebufferingrelayunitimmediately.Furthermore,themonitorordersthebufferingrelayunittotransmitallofthedatatothebackupsiteafteraconstantnumberoftransmittingdataupdate.Then,weformulatethestochasticmodelasfollows:(1)Aclientrequeststhedataupdatetothestorage,anditstimehasanexponentialdistribution(1−e−αt).TheserverinthemainsiteupdatesMonitorMainsiteBackupsiteBufferingServerStorageRelayunitStorageServerClientFig.4Outlineofaserversystem. 183September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisofaServerSystemwithReplicationSchemes169thestoragedatabase,andtransmitstheupdateddataandtheaddressofthephysicallocationupdateddataonthestorage.Thetotalofdataupdate,thetransmissionofthedataandtheaddressofthelocationupdateddatarequiresthetimeaccordingtoanexponentialdistribution(1−e−βt).(2)Themonitorordersthebufferingrelayunittotransmitallofthedatatothebackupsite,aftertheserverinthemainsiteupdatesthedatabyrequestingfromclientandtransmitsthedataatn(n=1,2,···)times(replication).(a)Thereplicationtimehasanexponentialdistribution(1−e−wt).(b)Ifadisasteroccursinthemainsitewhilethereplicationisexecuted,themonitorimmediatelyorderstomigratetheroutineworkfromthemainsitetothebackupsite.(3)Whenadisasterhasoccurredinthemainsite,theroutineworkismigratedfromthemainsitetothebackupsite,andtheserverperformstheroutineworkinthebackupsite(systemmigration).TheserversystemisshowninFig.5:(a)Adisasteroccursaccordingtoanexponentialdistribution(1−e−λt).Adisasterdoesnotoccurinthemonitor,thebufferingrelayunitandthebackupsite.BackupSiteTableofAddressheldinBufferingUnitBufferingStorageServerRelayUnitTableofAddressheldinStorageClientFig.5Outlineofaserversystemaftersystemmigration. 184September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book170ReliabilityModelingwithApplications(b)Whentheroutineworkismigratedfromthemainsitetothebackupsite,themonitorordersthebufferingrelayunittotransfertheaddressofupdateddataheldinittothebackupsite.Thesys-temmigrationexecutesaccordingtoanexponentialdistribution(1−e−gt).(i)Whenaclientrequeststhedataupdateorthedataread,theserverconfirmsthetableofaddressheldinthebufferingrelayunit.Iftheaddressoftherequesteddatacorrespondswiththeaddressinthetable,itistransmittedfromthebufferingrelayunittothebackupsiteinstantlyandthetableoftheaddressheldinthestorageisupdated.(ii)Therequesttimehasanexponentialdistribution(1−e−α1t).Theconfirmationoftheaddressheldinthebufferingrelayunitrequiresthetimeaccordingtoanexponentialdistribution(1−e−ht).(iii)TheprobabilitythattheaddressoftherequesteddatacorrespondswiththeaddressheldinthebufferingrelayunitisP(n)(n=1,2,···).Wherenrepresentsthenumberoftransmittingupdateddatainthebufferingrelayunit.Thetransmissionofdataanddataupdaterequiresthetimeaccordingtoanexponentialdistribution(1−e−β1t).(iv)Theprobabilitythattheaddressoftherequesteddatadoesnotcorrespondwiththeaddressheldinthebufferingrelayunitis1−P(n)(n=1,2,···).Thedataupdateorthedatareadrequirethetimeaccordingtoanexponentialdistribution(1−e−β2t).Undertheaboveassumptions,wedefinethefollowingstatesoftheserversystem:State0:Systembeginstooperateorrestart.Statesa1,a2,···,an:Whenaclientrequeststhedataupdatetothestor-age,thedataupdatebegins.Statesb1,b2,···,bn−1:Dataupdateandthetransmissionofthedataandtheaddressofthelocationupdateddataiscompleted.Statebn:Whentheserverinthemainsitehastransmittedthedataandtheaddressofthelocationupdateddataatn(n=1,2,···)times,replicationbegins.StatesF0,F1,···,Fn:Disasteroccursinthemainsite.States00,01,···,0n:Routineworkismigratedfromthemainsitetothebackupsite,andtheserverperformstheroutineworkinthebackupsiteandthebufferingrelayunit. 185September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisofaServerSystemwithReplicationSchemes171a1a2....anb1b2....bn0F0F1....Fn-1Fn0001....0n-10nd0d1e1...dn-1en-1dnenSFig.6Transitiondiagrambetweenasystemstates.Statese1,e2,···en:Whenaclientrequeststhedataupdate,thedataup-datebeginsinthebufferingrelayunit.Statesd1,d2,···dn:Whenaclientrequeststhedataupdate,thedataup-datebeginsinthestorageatthebackupsite.StateS:Dataupdateiscompleted.ThesystemstatesdefinedaboveformaMarkovrenewalprocess[Osaki(1992);Yasui,NakagawaandSandoh(2002)],whereSisanabsorbingstate.AtransitiondiagrambetweensystemstatesisshowninFig.6.TheLSTransformsoftransitionprobabilitiesQj,k(t)(j=0,ai,bi(i=1,2,···,n);k=0,ai,bi,F0,Fi(i=1,2,···,n))aregivenbythefollowingequations:∗∗α∗∗λQ0,a1(s)=Qbi−1,ai(s)=s+α+λ,Q0,F0(s)=Qbi,Fi(s)=s+α+λ,∗β∗λ∗wQai,bi(s)=s+β+λ,Qai,Fi−1(s)=s+β+λ,Qbn,0(s)=s+w+λ,∗λ∗∗gQbn,Fn(s)=,QF0,00(s)=QFi,0i(s)=,s+w+λs+g∗α1h∗α1[1−P(i)]hQ00,d0(s)=s+αs+h,Q0i,di(s)=s+αs+h,11∗α1P(i)h∗β1∗β2Q0i,ei(s)=,Qdi,S(s)=,Qei,S(s)=,s+α1s+hs+β1s+β2(i=1,2,···,n), 186September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book172ReliabilityModelingwithApplicationswhereb0≡0.WederivetheexpectednumberMB(n)oftheupdateddatainbufferingrelayunituntilstateS.TheLStransformsP∗(s)(i=0,Fi,ei,S1,2,···,n)oftheprobabilitydistributionsP0,Fi,ei,S(t)(i=1,2,···,n)fromstate0tostateSthroughstateFi,ei(i=1,2,···,n)untiltimetareΠiQ∗(s)Q∗(s)[Q∗(s)+Q∗(s)Q∗(s)]j=1bj−1,ajaj,bjbi,Fibi,ai+1ai+1,Fi×Q∗(s)Q∗(s)Q∗(s)∗Fi,0i0i,eiei,SP0,Fi,ei,S(s)=1−Q∗(s)ΠnQ∗(s)Q∗(s),0,a1j=1aj,bjbn,0(i=1,2,···n−1),(17)Q∗(s)ΠnQ∗(s)Q∗(s)Q∗(s)Q∗(s)Q∗(s)∗0,a1i=1ai,bibn,FnFn,0n0n,enen,SP0,Fn,en,S(s)=1−Q∗(s)ΠnQ∗(s)Q∗(s),0,a1i=1ai,bibn,0(18)whereb0≡0.Hence,theprobabilityP0,Fi,ei,S(i=1,2,···,n)fromstate0tostateFi(i=1,2,···,n)throughstateFi,ei(i=1,2,···,n)isderivedbyP≡lim[P∗(s)](i=1,2,···,n).Thereafter,theexpected0,Fi,ei,S0,Fi,ei,Ss→0numberMB(n)untilstateSis∑nY(n)MB(n)≡iP0,Fi,ei,S=1−wXn,(19)i=1w+λwhereαβX≡,(α+λ)(β+λ)n∑−1inλnY(n)≡(1−X)iXP(i)+XP(n).λ+wi=1Similarly,wederivetheexpectednumberMR(n)ofreplication.TheLStransformM∗(s)oftheexpectednumberdistributionM(t)in[0,t]areRRQ∗(s)Πn−1Q∗(s)Q∗(s)Q∗(s)Q∗(s)∗0,a1i=1ai,bibi,ai+1an,bnbn,0MR(s)=1−Q∗(s)Πn−1Q∗(s)Q∗(s)Q∗(s)Q∗(s),0,a1i=1ai,bibi,ai+1an,bnbn,0()[]nwαβs+w+λ(s+α+λ)(s+β+λ)=()[]n.1−wαβs+w+λ(s+α+λ)(s+β+λ) 187September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisofaServerSystemwithReplicationSchemes173Hence,theexpectednumberMR(n)iswXn∗w+λMR(n)≡lim[MR(s)]=wn,(20)s→01−Xw+λ3.2OptimalPolicyWeproposeanoptimalpolicytoreducewasteofcostsforreplicationandtransmittingtheupdateddatainbufferingrelayunit.Thatis,wecalculatetheexpectedcostuntilstateSandderiveanoptimalreplicationintervaln∗tominimizeit.LetcbethecostfortransmittingtheupdateddataBinbufferingrelayunitandcRthecostforreplication.Then,wedefinethecostC(n)asfollows:C(n)≡cRMR(n)+cBMB(n)(n=1,2,···).(21)Weseekanoptimalreplicationintervaln∗(1≤n∗≤∞)whichmini-mizesC(n)in(21).FromC(n+1)−C(n)≥0,()Y(n+1)−Y(n)−Xnw[Y(n+1)−XY(n)]λ+wcR()≥(n=1,2,···).w(1−X)XncBλ+w(22)Denotingtheleft-handsideof(22)byL(n),L(n)−L(n−1)wn{}1−Xλ+w=Y(n+1)−Y(n)−X[Y(n)−Y(n−1)],w(1−X)Xnλ+wThebracketisY(n+1)−Y(n)−X[Y(n)−Y(n−1)]{λXn+1w(α+β+λ)−αβ=[nP(n)−(n−1)P(n−1)]λ+wαβ}+(n+1)P(n+1)−nP(n),Clearly,L(∞)=∞.Thus,ifw(α+β+λ)−αβ>0andnP(n)isaincreasingfunctionofnthenthereexistsafiniteanduniqueminimumn∗(1≤n∗<∞)whichsatisfies(22). 188September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book174ReliabilityModelingwithApplicationsInparticular,whenP(n)=p,L(n)increasesstrictlywithnto∞,andhence,afiniten∗alwaysexists.Next,weconsidertheparticularcasethatP(n)≡1−e−γn(n=1,2,···).Then,nP(n)−(n−1)P(n−1)=(1−e−γ)ne−γ(n−1)+1−e−γ(n−1)>0,Therefore,ifw(α+β+λ)−αβ>0,nP(n)isaincreasingfunctionofnandthereexistsafiniteanduniquen∗whichsatisfies(22).Example9.2.Wecomputenumericallyanoptimalreplicationintervaln∗whichmin-imizesC(n)in(21).Supposethatthemeantime1/βrequiredforthedataupdateisaunittime.Itisassumedthatthemeangenerationin-tervalofrequestthedataupdateis(1/α)/(1/β)=10∼40,thera-tiothatrequesteddatacorrespondswiththedataheldinthebufferingrelayunitis1/γ=5,50,themeangenerationintervalofadisasteris(1/λ)/(1/β)=1000,5000,themeantimerequiredforthereplicationis(1/w)/(1/β)=40,80,Further,weintroducethefollowingcosts:ThecostfortransmittingthedataandtheaddressofthelocationupdateddataiscB=1,thecostforreplicationiscR/cB=5,10.Table3givestheoptimalreplicationintervaln∗whichminimizesthecostC(n).Forexample,whencB=1,cR=5and1/β=1,1/γ=5,1/α=10,1/w=40,1/λ=1000,theoptimalreplicationintervalisn∗=27.Thisindicatesthatn∗increasewith1/γ,1/λandc/c.Thus,weshouldRBholdmoreamountofupdateddatainthebufferingrelayunitthanexecutereplication.Moreover,n∗decreasewith1/αand1/w.Inthiscase,weshouldexecutereplicationratherthanholdlargeamountsofupdateddatainit.Table3OptimalreplicationintervalntominimizeC(n)β/λ100050001/γcR/cBβ/wβ/α1020401020405402720166548355802318156146341040423224947051803830239168505403530256853425080322824645141104048403494735780453833907156 189September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisofaServerSystemwithReplicationSchemes1754ConclusionsThischapterhasstudiedanalyticallythetwostochasticmodelsofaserversystembyapplyingreplicationusingjournalingfilesandreplicationbuffer-ingrelaymethod.Further,wehavederivedthereliabilitymeasuresbyusingthetheoryofMarkovrenewalprocesses,andhavediscussedtheop-timalpolicywhichminimizestheexpectedcost.Finally,wehavegiventhenumericalexamplesofeachmodelinordertounderstandtheresultseasily,andhaveevaluatedthemundersomestandardparameters.ReferencesFujita,T.andYata,K.(2005).Asynchronousremotemirroringwithjournalingfilesystems,Trans.IPSJapan,46,SIG16,pp.56–68.Goda,K.andKitsuregawa,M.(2007).Astudyonpowerreductionmethodofdiskstorageforlogforwardingbaseddisasterrecoverysystems,DBSJLetters,6,1,pp.69–72.Imai,T.,Araki,S.,Sugiura,T.,Fujita,N.andSuemura,Y.(2004).Session-uninterruptedDisasterRecoveryacrossDistributedDataCentersIEICETechnicalReport,NS2003-293,IN2003-248,pp.199–202.Kan,M.,Yamato,J.,Kaneko,Y.andKikuchi,Y.(2005).Anapproachofshort-eningtherecoverytimeinthereplicationbufferingrelaymethodfordisasterrecovery,IEICETechnicalReport,CPSY2005-19,pp.25–30.Kimura,M.,Imaizumi,M.andNakagawa,T.(2010).OptimalReplicationIn-tervalofanAsynchronousReplicationusingJournalingFiles,inThe4thAsia-PacificInternationalSymposiumonAdvancedReliabilityandMain-tenanceModeling(Wellington,NewZealand),pp.349–356.Kimura,M.,Imaizumi,M.andNakagawa,T.(2011).ReliabilityConsiderationofaReplicationwithLimitednumberofJournalingFiles,inThe7thIn-ternationalConferenceon“MathematicalMethodsinReliability”:TheoryMethodsApplications(Beijing,China),pp.948–953.Kimura,M.,Imaizumi,M.andNakagawa,T.(2011).ReliabilityConsiderationofaServerSystemwithReplicationBufferingRelayMethodforDisas-terRecovery,inInternationalConferencesASEA/DRBC/EL,Communi-cationsinComputerandInformationScience257(JejuIsland,Korea),pp.392–398.Kimura,M.,Imaizumi,M.andNakagawa,T.(2012).ReliabilityModelingforaServerSystemwithBufferingRelayMethod,inThe5thAsia-PacificIn-ternationalSymposiumonAdvancedReliabilityandMaintenanceModelingV(Nanjing,China),pp.255–262.Nakamura,N.,Fujiyama,K.,Kawai,E.andSunahara,H.(2007).AFlexibleReplicationMechanismwithExtendedDatabaseConnectionLayersforDis-asterRecoverySystem,Trans.IPSJapan,48,pp.1562–1572.Osaki,S.,(1992).AppliedStochasticSystemModeling(Springer-Verlag,Berlin). 190September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book176ReliabilityModelingwithApplicationsVERITASSoftwarecorporation.VERITASvolumereplicationforUNIXdatasheet,http://eval.veritas.com/mktginfo/products/Datasheets/HighAvailability/vvrdatasheetunix.pdf.Watabe,S.,Suzuki,Y.,Mizuno,K.andFujiwara,S.(2007).Evaluationofspeed-upmethodsofdatabaseredoprocessingforlog-baseddisasterrecoverysys-tems,DBSJLetters,6,1,pp.133–136.Yamato,J.,Kan,M.andKikuchi,Y.(2006).Storagebaseddataprotectionfordisasterrecovery,J.IEICE,89,9,pp.801–805.Yasui,K.,Nakagawa,T.andSandoh,H.,(2002).Reliabilitymodelsindatacommunicationsystems,StochasticModelsinReliabilityandMaintenance,(editedbyS.Osaki)(Springer-Verlag,Berlin),pp.281–301. 191September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookPART3ComputerSystems177 192May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank 193September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookChapter10Two-DimensionalSoftwareReliabilityGrowthModelsShinjiInoueandShigeruYamadaDepartmentofSocialManagementEngineering,TottoriUniversity,Minami4-101,Koyama,Tottori-shi680-8552,Japan1IntroductionSoftwarereliabilityassessmentinatesting-phaselocatedinafinalstageofthesoftwaredevelopmentprocessisoneoftheimportantactivitiesindevelopingahighly-reliablesoftwaresystem.Inthetesting-phase,anim-plementedsoftwaresystemistestedtodetectandcorrectsoftwarefaultslatentinthesoftwaresystem.Thesoftwaredevelopmentmanagerhastoassessthesoftwarereliabilityofthefinalsoftwareproductespeciallyinthefinalphaseofthesoftwaredevelopmentprocessforshippingareliablesoftwaresystemtotheuser.Asoftwarereliabilitygrowthmodel(abbre-viatedasSRGM)[Musa(1972);YamadaandOsaki(1985);Pham(2003)]isknownasoneofthefundamentaltechnologiesforquantitativesoftwarereliabilityassessment,andplaysanimportantroleinsoftwareprojectman-agementforproducingahighly-reliablesoftwaresystem.TheSRGMisamathematicalmodel,whichdescribesasoftwarereliabilitygrowthprocessinatestingphaseofthesoftwaredevelopmentprocessandtheoperationalphasebyregardingthenumberofsoftwarefaultsdetectedduringacertaintime-intervalorthesoftwarefailure-occurrencetimeintervalastherandomvariables.AfterdescribingthesoftwarereliabilitygrowthprocessbyusingtheSRGM,weassesssoftwarereliabilityquantitativelybasedonthesoft-warereliabilityassessmentmeasures,whicharederivedfromtheSRGMappliedforthesoftwarereliabilityanalysis.179 194September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book180ReliabilityModelingwithApplicationsMostofSRGMsbeingclassifiedwithsoftwarefailure-detectioncountmodelsaredevelopedunderthebasicassumptionthatasoftwarereliabilitygrowthprocessdependsonlyonthetesting-timeortheoperational-time,suchascalendertime.However,itisdifficulttosaythatthereliabilityofthesoftwaresystemdependsonlythetesting-timedurationsinceitisknownthatthesoftwarereliabilitygrowthprocessdependsnotonlyonthetesting-timebutalsotheseveraltesting-effortfactorswhicharerelatedtothesoftwarereliabilitygrowthprocess,suchastest-executiontime(CPUhours)[Yamadaetal.(1986)],testing-skilloftestengineers[FujiwaraandYamada(2001)],andtesting-coverage[InoueandYamada(2004)].Underthebackgroundmentionedabove,two-dimensional(orbivariate)softwarereliabilitygrowthmodelingapproacheshavebeendiscussedinrecentyears.Forexamples,IshiiandDohi[IshiiandDohi(2006)]proposedsoftwarere-liabilitymodelingframeworkbasedonatwo-dimensionalnonhomogeneousPoissonprocess(abbreviatedasNHPP),wheretheyassumedthatthesoft-warereliabilitygrowthprocessdependsonsimultaneouscalendertimeandCPUhours.Ishiietal.[Ishiietal.(2008)]proposedatwo-dimensionalgeometricsoftwarereliabilitymodelbytakingaccountofcalendertimeandthenumberofexecutedtestcases.Wediscussothertwo-dimensionalsoftwarereliabilitygrowthmodelingapproaches,inwhichthesoftwarereliabilitygrowthprocessdependsonthetesting-timeandthetesting-effortfactors.Especially,ourmodelingapproachesconsiderwiththedegreeoftheimpactofthethesetwofac-torstothesoftwarereliabilitygrowthprocessandtheeffectofthepro-gramsizetothereliabilitygrowthprocess,respectively.Therefore,wecanexpectmorefeasiblesoftwarereliabilitymeasurementandassessmentbyourtwo-dimensionalsoftwarereliabilitygrowthmodelingapproaches.Wedefinesomestochasticquantitiesfordescribingthetwo-dimensionalsoft-warefailure-occurrenceorfault-detectionphenomenonbeforediscussingourtwo-dimensionalsoftwarereliabilitygrowthmodelingframeworks.Af-terthat,two-typesoftwo-dimensionalsoftwarereliabilitygrowthmodelingapproachesarediscussed.Then,comparingwithexistingone-dimensionalSRGMs,wecheckperformanceofourmodelsintermsofagoodness-of-fitcriterion,andshowexamplesoftheapplicationoftwo-dimensionalsoftwarereliabilityanalysisbasedonourtwo-dimensionalSRGMdevelopedbyourmodelingframeworkbyusingactualfaultcountdata. 195September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookTwo-DimensionalSoftwareReliabilityGrowthModels181Fig.1Stochasticquantitiesforthetwo-dimensionalsoftwarefailure-occurrenceorfault-detectionphenomenon.2StochasticQuantitiesforTwo-DimensionalModelingOurtwo-dimensionalSRGMdescribesasoftwarereliabilitygrowthprocessdependingonthefollowingtwo-typesofsoftwarereliabilityfactors:Testing-timeandtesting-effortfactors.Figure1illustratestherelatedstochas-ticquantitiesforthesoftwarefailure-occurrencephenomenononthetwo-dimensionalspaceconsistingofthesetwosoftwarereliabilitygrowthfactors.AccordingtoFig.1,thestochasticquantitiesaredefinedasfollows:N(s,u)isatwo-dimensionalstochasticprocessrepresentingthenumberoffaultsdetecteduptotesting-timesandtheamountoftesting-effortexpendituresu,Skthek-thsoftwarefailure-occurrencetime(k=0,1,2,···;S0=0),Ukthetesting-effortexpendeduptothek-thsoftwarefailure-occurrence(k=0,1,2,···;U0=0),Xithetime-intervalbetweenthe(i−1)-standthei-thsoftwarefailure-occurrences(i=1,2,···;X0=0),andYithetesting- 196September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book182ReliabilityModelingwithApplicationseffortexpendedduringbetweenthe(i−1)-standthei-thsoftwarefailure-occurrences(i=1,2,···;Y0=0).Inthesestochasticquantitiesabove,it∑k∑kisapparentthatSk=i=1XiandUk=i=1Yi,whereXi=Si−Si−1andYi=Ui−Ui−1.ThefaultcountdatawhichconsistsofNdatapairs,(si,ui,yi)(i=0,1,2,···,N;s0 197September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookTwo-DimensionalSoftwareReliabilityGrowthModels183NHPPisgivenas{H(s,u)}nPr{N(s,u)=n}=exp[−H(s,u)]n!(n=0,1,2,···),(2)∫s∫uH(s,u)=h(τ,υ)dτdυ00whereH(s,u)andh(τ,υ)representameanvaluefunctionandtheintensityfunctionofthetwo-dimensionalNHPP,respectively.Asoneoftheexamples,weapplysuchapproachinEqs.(1)and(2)tothefollowingone-dimensionalWeibull-typeSRGM[KeillerandMiller(1991)]:H(t)≡γ(t)()βt=(0<β<1;ρ>0),(3)ρwhereH(t)isameanvaluefunctionofaone-dimensionalNHPP[Naka-gawa(2005)],βthesoftwarereliabilitygrowthparameter,andρthescaleparameter.Ifweapplyourapproachtotheone-dimensionalWeibull-typeSRGM,thenwecanextendthemeanvaluefunctioninEq.(3)as()βsαu1−αH(s,u)≡γ(s,u)=.(4)ρEquation(4)hasthefollowingproperties:whenα=1,Eq.(4)canberegardedastheconventional(orone-dimensional)Weibull-typeSRGM,inwhichthesoftwarereliabilitygrowthprocessdependsonlyonthetesting-time.Ontheotherhand,Eq.(4)becomesatesting-effortdependentSRGMwhenα=0.Aswementionedabove,ourapproachkeepsconsistencywithone-dimensionalSRGMs.Wecallthistwo-dimensionalSRGMinEq.(4)“OurModel1”.3.2BinomialTwo-DimensionalSRGMWediscussanotherapproachfortwo-dimensionalsoftwarereliabilitygrowthmodeling.Ouranotherapproachisbasedonthefollowingbasicassumptions[IshiiandDohi(2006)]:(A1)Wheneverasoftwarefailureisobserved,thefaultisdetectedim-mediately,andnonewfaultsareintroducedinthefault-detectionprocedures. 198September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book184ReliabilityModelingwithApplications(A2)Eachsoftwarefailureoccursatindependentlyandidenticallydis-tributedrandomtimewiththebivariateprobabilitydistribution∫s∫ufunctionF(s,u)≡Pr{S≤s,U≤u}=f(x,y)dxdy,where00f(x,y)andPr{A}representthejointdensityfunctionandtheprob-abilityofeventA,respectively.(A3)Theinitialnumberoffaultsinthesoftwaresystem,N0(>0),isarandomvariable,andisfinite.Fromthebasicassumptionsabove,wedevelopatwo-dimensionalsoftwarereliabilitygrowthmodelingframeworkwithprogramsize.Theprogramsizeisoneofthemetricsofsoftwarecomplexity,whichinfluencesthesoft-warereliabilitygrowthprocess.Fromthebasicassumptions,wehavetheprobabilitymassfunctionofN(s,u)as()∑nPr{N(s,u)=m}={F(s,u)}m{1−F(s,u)}n−mmn×Pr{N0=n}(m=0,1,2,···).(5)Wenotethattherangeofncannotbefixedbecausetheprobabilitydistri-butionfunctionoftheinitialfaultcontentN0isnotstillgiveninEq.(5).Inourmodelingapproach,weassumethefollowingbinomialdistributionwithparameter(K,λ)fortheprobabilitymassfunctionoftheinitialfaultcontentinEq.(5)toincorporatetheeffectoftheprogramsizeonthetwo-dimensionalsoftwarereliabilitygrowthprocess:()KnK−nPr{N0=n}=λ(1−λ)n(0<λ<1;n=0,1,···,K).(6)Equation(6)hasthefollowingphysicalassumptionsfortheinitialfaultcontent[Kimuraetal.(1993);InoueandYamada(2007)]:(a)ThesoftwaresystemconsistsofKlinesofcode(LOC)atthebeginningofthetesting-phase.(b)Eachcodehasafaultwiththeconstantprobabilityλ.(c)Eachsoftwarefailurecausedbyafaultremaininginthesoftwaresystemoccursindependentlyandrandomly.SubstitutingEq.(6)intoEq.(5),wehave()KmK−mPr{N(s,u)=m}={λF(s,u)}{1−λF(s,u)}m(0<λ<1;m=0,1,2,···K).(7) 199September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookTwo-DimensionalSoftwareReliabilityGrowthModels185FromEq.(7),wecanseethatseveraltypesoftwo-dimensionalSRGMswiththeeffectoftheprogramsizecanbedevelopedbyassumingasuitablebivariatesoftwarefailure-occurrencetimedistribution,F(s,u).Asoneoftheexamples,ifweassumethateachsoftwarefailure-occurrencetimefollowsthefollowingbivariateprobabilitydistributionfunc-tionproposedbyGumbel[Gumbel(1960)]:F(s,u)=(1−e−as)(1−e−bu)(1+ze−as−bu)(a>0,b>0,−1≤z≤1),(8)thentheexpectationofN(s,u)isderivedasE[N(s,u)]=KλF(s,u)=Kλ(1−e−as)(1−e−bu)(1+ze−as−bu).(9)Wecallthetwo-dimensionalSRGMinEq.(9)“OurModel2”,whichisdevelopedbyourmodelingapproachinEq.(7).4ParameterEstimationWediscussparameterestimationmethodsfortwo-dimensionalSRGMsinEqs.(4)and(9).SupposethatNdatapairshavebeenobserved.WenowdiscussaparameterestimationmethodforOurModel1inEq.(4).InOurModel1,weneedtoestimateparameters,α,β,andρ,respectively.Theparameterestimatesofthistwo-dimensionalSRGMcanbeeasilyobtainedbyusingthemultipleregressionanalysissincewecanderivethefollowingequationfromEq.(4)aslogγ(s,u)=−βlogρ+αβlogs+(1−α)βlogu,(10)bytakingthenaturallogarithmofEq.(4).Then,wehavethefollowingmultipleregressionequation:Yi=a0+a1Ki+a2Li+ϵi,(11)whereYi=logyi,Ki=logsi,Li=logui,(12)a0=−βlogρ,a1=αβ,a2=(1−α)β. 200September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book186ReliabilityModelingwithApplicationsInEq.(11),ϵiisastandardnormalerrortermwithhomoscedasticity,i.e.,equalityofvariance.FromEqs.(11)and(12),thesumofthesquaredverticaldistancesfromtheactualdatapointstothepresumedpolynomial,S(a0,a1,a2),isderivedas∑NS(a,a,a)=ϵ2012ii=1∑N2={Yi−(a0+a1Ki+a2Li)}.(13)i=1Parameterestimates,ba0,ba1,andba2,oftheparametersa0,a1,anda2areestimatedbyminimizingEq.(13).Thatis,solvingthesimultaneousequa-///tions,∂S∂a0=∂S∂a1=∂S∂a2=0,yieldstheparameterestimatesofa0,a1,anda2,respectively.Finally,wecanobtainparameterestimatesαb,βb,andρboftheparameterα,β,andρasba1αb=,ba1+ba2βb=ba1+ba2,(14)[]ba0ρb=exp−,ba1+ba2byusingtheestimatedpartialregressioncoefficients,ba0,ba1,andba2.AndwealsodiscussaparameterestimationmethodforOurModel2inEq.(9).InEq.(9),consideringthatinformationoftheprogramsizecanbeeasilyobtainedfromanactualsoftwareproject,weneedtoestimateparameters,λ,a,b,andz,respectively.Wecanestimatetheparametersofthistwo-dimensionalSRGMbyusingthemethodofmaximum-likelihood.Now,wederivethelikelihoodfunction,l,forthetwo-dimensionalstochasticprocess,N(s,u),inEq.(5).Basedonthebasicassumptionsin3.2,wehavethelikelihoodfunctionasl=Pr{N(s1,u1)=y1,N(s2,u2)=y2,···,N(sN,uN)=yN}∏N=Pr{N(si,ui)=yi|N(si−1,ui−1)=yi−1}i=1·Pr{N(s1,u1)=y1}.(15) 201September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookTwo-DimensionalSoftwareReliabilityGrowthModels187byusingtheBayes’formulaandMarkovproperty.Theconditionalproba-bilityinEq.(15)canbederivedasPr{N(si,ui)=yi|N(si−1,ui−1)=yi−1}()K−yi−1yi−yi−1K−yi={z(Φi−1,Φi)}{1−z(Φi−1,Φi)},(16)yi−yi−1whereλ{F(si,ui)−F(si−1,ui−1)}z(Φi−1,Φi)=.(17)1−λF(si−1,ui−1)ByusingEqs.(16)and(17),wecanrewriteEq.(15)asN()∏K−yi−1yi−yi−1K−yil={z(Φi−1,Φi)}{1−z(Φi−1,Φi)},(18)yi−yi−1i=1whereF(s0,u0)=F(s0,u)=F(s,u0)=0.Basedonthelogarithmiclike-lihoodfunctionofEq.(18),wecanderivethesimultaneouslogarithmiclikelihoodequationswithrespecttoeachparameterofthetwo-dimensionalSRGMinEq.(9).Accordingly,wecanobtainmaximum-likelihoodesti-matesoftheparametersbysolvingthesimultaneouslogarithmiclikelihoodequationsnumerically.5ModelComparisonWeconductmodelcomparisonofourtwo-dimensionalSRGMswiththefollowingexistingone-dimensionalSRGMs:One-dimensionalWeibull-type(called“ONE-DWEIBULL”)inEq.(3),atesting-coveragedepen-dentSRGM(“TCD”)[InoueandYamada(2004)],anexponentialSRGM(“EXPO”)[GoelandOkumoto(1979)],andadelayedS-shapedSRGM(“DELAYED-S”)[Yamadaetal.(1983)].Thefollowinghasmorede-tailsonthetesting-coveragedependent,exponential,anddelayedS-shapedSRGMs:•Testing-CoverageDependentSRGM[InoueandYamada(2004)]:ThisSRGMisdevelopedbyconsideringtherelationshipbetweenthetesting-coverageattainmentandthenumberofdetectedfaults.LetHTC(t)betheexpectednumberofthefaultsdetecteduptotesting-timetandthemeanvaluefunctionoftheone-dimensionalNHPP. 202September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book188ReliabilityModelingwithApplicationsThen,wehave([∫t])HTC(t)=a1−exp−sc(x)dx0(a>0,0 203September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookTwo-DimensionalSoftwareReliabilityGrowthModels189Table1ResultofmodelcomparisonbasedonMSE.DS1DS2OURMODEL11.963563.09947OURMODEL254.753575.0829ONE-DWEIBULL1425.24205.688TCD111.83849.9692EXPO603.19873.0885DELAYED-S150.00536.6406HDS(t)betheexpectednumberoffaultsdetecteduptotesting-timet.Then,[]H(t)=a1−(1+bt)e−bt(a>0,b>0),(22)DSwherearepresentstheexpectedinitialfaultcontentandbthefailure-occurrencerateorthefault-detectionrate.Theparametersareesti-matedbythemethodofmaximum-likelihood.Weusetwoactualdatasets,DS1andDS2[FujiwaraandYamada(2002)].DS1consistsof24datapairs:(sk,uk,yk)(k=1,2,···,24;s24=24(weeks),u=0.9095,y=296,K=1.972×105(LOC))andDS2242422datapairs:(sk,uk,yk)(k=1,2,···,22;s22=22(weeks),u22=0.9198,y=212,K=1.630×105(LOC)),whereu,thetesting-effort24ifactor,representsthetesting-coverageattaineduptotesting-timesi.Nowweconductmodelcomparisonbasedonthewell-knownmeansquareerrors(abbreviatedasMSE).TheMSEiscalculatedbydividingthesumofsquaredverticaldistancebetweentheobservedandestimatedcu-mulativenumberoffaults,yiandyb(Ψi),detectedduringthetime-interval(0,Ψi],respectively,bythenumberofobserveddatapairs.Ψiisthesoft-warereliabilitygrowthfactorineachmodel.TheMSEiscalculatedas∑N12MSE={yi−yb(Ψi)}.(23)Ni=1FromEq.(23),thesmallervalueofMSErepresentsfittingbettertotheactualdata.Table1showstheresultsofmodelcomparisonsbasedonMSE.FromTable1,wecanseethatourtwo-dimensionalSRGMshavebetterperformanceintermsofMSEamongSRGMscitedinthismodelcomparison.Especially,OurModel1hasthebestperformance.And,wecansaythattwo-dimensionalsoftwarereliabilitymeasurement,inwhichitisassumedthatthesoftwarereliabilitygrowthprocessdependsnotonlyontesting-timebutalsotesting-effortfactors,isusefulandfeasibleforactualassessmentofsoftwarereliability. 204September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book190ReliabilityModelingwithApplications4003503002502001501001500.8CumulativeNumberofDetectedFaults00.6050.4age10verTestingT150.2-Coime(num20berofweeks25300Testing)Fig.2Estimatedtwo-dimensionalmeanvaluefunction,γb(s,u)(DS1).6NumericalExamplesWeshowexamplesofapplicationofOurModel1inEq.(4),whichhasthebestperformanceinourmodelcomparisonintermsofMSE,byDS1.FromDS1,wecanestimatetheparameters,α,β,andρ,asαb=0.17555,βb=0.95652,andρb=0.00415,respectively,followingtotheparameterestimationmethoddiscussedin4.Figure2depictsthebehavioroftheestimatedtwo-dimensionalmeanvaluefunctioninEq.(4),whichdependsonthetwo-dimensionalspace.InFig.2,thedottedlinesandthecurvedsurfacerepresentthebehavioroftheactualdataandtheestimatedbehavioroftheexpectednumberofdetectedfaults,respectively.FromFig.2,wecanseethattheestimatedtwo-dimensionalSRGMcandescribetheactualphenomenonthatthesoft-warereliabilitygrowthdoesnotobservedevenifalotoftesting-timeisexpendedunderthesituationthattheamountoftesting-effortexpendi-turesisnotincreased.Suchactualphenomenoncannotbedescribedbyone-dimensionalSRGMs.Wealsoshowanumericalexampleofanoperationalsoftwarereliabilitybasedontheestimatedtwo-dimensionalmeanvaluefunctioninFig.2.Theoperationalsoftwarereliability[IshiiandDohi(2006)]isdefinedastheprobabilitythatasoftwarefailuredoesnotoccurintheoperationaltime-interval(se,se+η](se≥0,η≥0)giventhattheamountofthetesting-effort 205September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookTwo-DimensionalSoftwareReliabilityGrowthModels191'ޓޓޓޓޓޓޓޓޓޓޓޓޓޓޓޓ(:SoftwareFailure-OccurrenceǯueTesting-EffortFactor㨪㨪㨪㨪seTesting-TimeFactorFig.3Illustrationforconceptofoperationalsoftwarereliability.expenditurehasbeentakinguptouebytesting-timese.Figure3showsanillustrationfortheconceptoftheoperationalsoftwarereliability.Therefore,fromthepropertiesofthetwo-dimensionalNHPP,theoperationalsoftwarereliabilityisformulatedas[{}]R(η|se,ue)=exp−H(se+η,ue|Θb)−H(se,ue|Θb),(24)whereΘbindicatesasetofparameterestimatesofatwo-dimensionalSRGM.Figure4showstheestimatedoperationalsoftwarereliabilityfunction,Rb(η|24,0.9095).FromFig.4,wecanestimatetheoperationalsoftwarereliability,Rb(1.0|24,0.9095),tobeabout0.127.7ConcludingRemarksFromtheviewpointofactualsoftwarefailure-occurrencephenomenon,itisnaturaltoconsiderthatasoftwarereliabilitygrowthprocessdepends 206September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book192ReliabilityModelingwithApplications10.90.80.70.60.50.40.3OperationalSoftwareReliability0.20.1000.30.60.91.21.51.82.12.42.73OperationTime(numberofweeks)Fig.4Estimatedoperationalsoftwarereliabilityfunction,Rb(1.0|24,0.9095)(DS1).notonlyonthetesting-timefactorbutalsothetesting-effortfactor,suchastest-executiontime,testing-skilloftestengineers,testing-coverage.Wediscussedtwo-typesoftwo-dimensionalsoftwarereliabilitygrowthmodelingframeworks,whichdescribeareliabilitygrowthprocessdependingontwo-typesofsoftwarereliabilitygrowthfactors:testing-timeandtesting-effortfactors.Andwedevelopedtwo-dimensionalSRGMsinourtwo-typesofthetwo-dimensionalmodelingframeworks.Afterthat,wediscussedparameterestimationmethodsofoureachmodelingframework,andcomparedper-formanceofourSRGMswithexistingone-dimensionalSRGMsbyusingfaultcountdataobservedinactualtestingphases.Byourmodelcompari-sonbasedonMSEandshowingnumericalexamplesforsoftwarereliabilityanalysisbasedonourtwo-dimensionalSRGMbyusingactualdata,wecansaythatourtwo-dimensionalsoftwarereliabilitymeasurementtechnologiesenableustoconductmorefeasiblesoftwarereliabilityanalysisthantheexistingsoftwarereliabilitymeasurementapproach,inwhichitisassumedthatthesoftwarereliabilitygrowthprocessdependsonlyontesting-time.Inourfuturestudies,weneedtoinvestigateusefulnessandvalidityofourtwo-dimensionalsoftwarereliabilitygrowthmodelingframeworksbyapplyingthemtomanyactualsoftwaredevelopmentprojectsandcompar-ingwithexistingone-andtwo-dimensionalSRGMs.Andregardingour 207September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookTwo-DimensionalSoftwareReliabilityGrowthModels193two-dimensionalsoftwarereliabilitygrowthmodelingframeworkinEq.(7),wehavetoresearchtheappropriaterangeofprogramsizebecausethestochasticpropertyontheinitialfaultcontent,abinomialdistribution,canberegardedasaPoissondistributionastheparameterKandλ,respec-tively,tendtotheinfiniteandzero.Furtherwealsoneedtoderivemoreusefulsoftwarereliabilityassessmentmeasuresbasedontwo-dimensionalmodelingframework.ReferencesAhn,C.W.,Chae,K.C.andClark,G.M.(1998).Estimatingparametersofthepowerlawprocesswithtwomeasuresoffailuretime,J.Qual.Tech.30,2,pp.127–132.Fujiwara,T.andYamada,S.(2001).Softwarereliabilitygrowthmodelingbasedontesting-skillcharacteristics:ModelandApplication,Elec.Commu.Japan—Part384,6,pp.42–49.Fujiwara,T.andYamada,S.(2002).C0coverage-measureandtesting-domainmetricsbasedonasoftwarereliabilitygrowthmodel,Intern.J.Rel.Quali.Safe.Eng.9,4,pp.329–340.Goel,A.L.andOkumoto,K.(1979).Time-dependenterror-detectionratemodelforsoftwarereliabilityandotherperformancemeasures,IEEETrans.Reliab.R-28,3,pp.206–211.Gumbel,E.J.(1960).Bivariateexponentialdistributions,J.ASA55,pp.698–707.Inoue,S.andYamada,S.(2004).Testing-coveragedependentsoftwarereliabilitygrowthmodeling,Intern.J.Reliab.Qual.Safe.Eng.11,4,pp.303–312.Inoue,S.andYamada,S.(2007).Discreteprogram-sizedependentsoftwarereli-abilityassessment:Modeling,estimation,andgoodness-of-fitcomparisons,IEICETrans.Fundamentals,E90-A,12,pp.2891–1609.Ishii,T.andDohi,T.(2006).Two-dimensionalsoftwarereliabilitymodelsandtheirapplication,Proc.12thPacificRimIntern.Symp.Depend.Comput.,pp.3-10.Ishii,T.,Fujiwara,T.andDohi,T.(2006).Bivariateextensionofsoftwarereli-abilitymodelingwithnumberoftestcases,Intern.J.Reliab.Qual.Safe.Eng.15,1,pp.1–17.Keiller,P.A.andMiller,D.R.(1991).Ontheuseandtheperformanceofsoftwarereliabilitygrowthmodels,Rel.Eng.Syst.Safe.32,1-2,pp.95–117.Kimura,M.,Yamada,S.,Tanaka,H.andOsaki,S.(1993).Softwarereliabilitymeasurementwithprior-informationoninitialfaultcontent,Trans.IPSJapan34,7,pp.1601–1609.Murthy,D.N.P.,Baik,J.,Wilson,R.J.andBulmer,M.R.(2006).Two-dimensionalfailuremodelinginSpringerHandbookofEngineeringStatistics(Pham,H.Ed.),Springer-Verlag,Berlin,pp.97–111. 208September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book194ReliabilityModelingwithApplicationsMurthy,D.N.P.,Iskandar,B.P.andWilson,R.J.(1995).Two-dimensionalfailure-freewarrantypolicies:Two-dimensionalpointprocessmodels,Opera.Res.,43,2,pp.356–366.Musa,J.D.,Iannio,D.andOkumoto,K.(1987).SoftwareReliabilityFMeasure-ment,Prediction,Application,McGraw-Hill,NewYork.Nakagawa,T.(2005).MaintenanceTheoryofReliability,Springer-Verlag,London.Pham,H.(2000).SoftwareReliability.Springer-Verlag,Singapore.Varian,H.R.(1991).IntermediateMicroeconomics—AModernApproach,(2ndEdition),W.W.Norton&Company,NewYork.Yamada,S.,Ohba,M.andOsaki,S.(1983).S-shapedreliabilitygrowthmodelingforsoftwareerrordetection,IEEETrans.Reliab.R-32,5,pp.475–478,485.Yamada,S.,Ohtera,S.andNarihisa,H.(1986).Softwarereliabilitygrowthmod-elswithtesting-effort,IEEETrans.Reliab.R-35,1,pp.19–23.Yamada,S.andOsaki,S.(1985).SoftwarereliabilitygrowthmodelingFModelsandapplications,IEEETrans.Soft.Eng.SE-11,12,pp.1431–1437. 209September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookChapter11HybridCoordinatedCheckpointingTechniqueUsingIncrementalSnapshotsMamoruOhara,MasayukiArai,SatoshiFukumotoandKazuhikoIwasakiFacultyofSystemDesign,TokyoMetoropolitanUniversity,6-6Asahigaoka,Hino,Tokyo191-0065,Japan1IntroductionIndistributedsystems,checkpointing/recoverymustmaintainconsistentsystemstates.Incoordinatedcheckpointing,processescoordinatewitheachotherinordertosaveaconsistentglobalstate.Becauseglobalstatessavedbycoordinatedcheckpointingarealwaysconsistent,recoveryopera-tionissimple.However,overheadforcoordinationissometimesunaccept-ablylargewhencommunicationcostishigh.Incontrast,uncoordinatedcheckpointingtechniquesattempttoreducecheckpointingoverheadbyin-dependentlycreatingcheckpointsineachprocess.Asaresult,recoverycostofuncoordinatedtechniquesisusuallyhigherthancoordinatedones.Wecanseeatrade-offrelationbetweencheckpointingoverheadinnormaloperationandrecoverycost[Tohma(1988)].Elnozahyetal.concludedthatrecentgreatreductionofcommuni-cationcostmakescoordinatedcheckpointingmostattractive[Elnozahyetal.(2002)].Largerecoverycostofuncoordinatedcheckpointingmaybeanobstacleintoday’sdistributedsystems.However,whenprocessescre-atecheckpointswithcoordinatedtechniques,theymustsuspendmessageexchangeandwaitforallprocesses.Therefore,whentherearerigorousrequirementsforrapidrecoveryafterafailure,thatis,asystemneedstocreatecheckpointsfrequentlyinordertodecreaserecoverycost,thesystemperformancemaybesignificantlydecreased.Itisnotacceptableinsystems195 210September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book196ReliabilityModelingwithApplicationsconsistingofnumerousprocessessuchasgridcomputingsystems[Camargoetal.(2004);KrishnanandGannon(2004);Wooetal.(2004)].Inthischapter,wepresentahybridC/Rtechnique,whichreducesfre-quencyofcoordinatedcheckpointingandstillrealizesrelativelyrapidre-covery.Intheproposedtechnique,thesystemperformscoordinatedcheck-pointingonlyataportionofperiodiccheckpointingopportunities.Attheothercheckpointingopportunities,processesindependentlysavetheirstatedifferentialsandmessagehistories,chagedorreceivedafterthelastcoordi-natedcheckpoint.Inthischapter,wecallthesavedinformationconsistingofthedifferentialsandthemessagehistoriesasanincrementalsnapshot.Bymergingcheckpointdatasavedbycoordinatedcheckpointingandtheincrementalsnapshots,wecanobtainthesameimageasthatobtainedbyuncoordinatedcheckpointing.Thatis,theproposedtechniqueisahybridtechniqueofcoordinatedanduncoordinatedC/Rforreducingcommunica-tioncost.Moreover,theproposedtechniquecanalsoreducetheoverheadforuncoordinatedcheckpointingbysubstitutingtheincrementalsnapshots.Intheproposedtechnique,thenumberofrecoverylinesisobviouslydecreasedcomparedtothatintraditionalcoordinatedtechniques.How-ever,wecanprobabilisticallyobtainadditionalrecoverylineswhichconsistofcoordinatedcheckpointsandtheincrementalsnapshots.Thus,wemaybeabletocreateamoreeffectivefault-tolerancemechanismthanthetra-ditionaltechniquesinthegross.Thatis,wehavepossibilitiesofimprov-ingrecoveryperformancewithoutadditionalcheckpointingoverhead,orinversely,fulfillingrestrictionsinrecoverycostwithlessstatesavingover-head.Moreover,processescandiscardallexistingcoordinatedcheckpointsandtheincrementalsnapshotswhentheycreateanewcoordinatedcheck-point,itiseasytoestimatestorageoverheadpriori.Thisisanadvantageoftheproposedtechniqueagainstthecommunication-inducedcheckpointing[ElnozahyandZwaenepoel(1992)].Inthischapter,weconstructasimpleanalyticalmodelforvalidatingeffectivenessoftheproposedtechnique.Weconstructmodelsestimatingstatesavingoverheadandrecoverycost,respectively.Inevaluationoftherecoverycost,wespeculatedtheprobabilitydistribution,whichexpressesmagnitudeofrollback,usingsimulations.Weexaminetwotechniquestogeneratetheincrementalsnapshots:thedividedsnapshottingtechnique,inwhichaprocesssavesdifferencesinitsstatebetweentwosuccessivesnapshots/checkpoints,andthecumulativesnapshottingtechnique,whereaprocessalwayssavesthedifferencesbetweenthelastcoordinatedcheckpointandthecurrentstate.Throughtheanalysesandnumericalexamples,we 211September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookHybridCoordinatedCheckpointingTechniqueUsingIncrementalSnapshots197foundthatwithlowmessagefrequency,theproposedtechniqueismoreeffectivethanthetraditionalcoordinatedtechnique.Comparingthetwosnapshottechniques,thecumulativeonewasmoreadvantageousinmanysituations.Notethattheanalyticalmodelspresentedinthischapterevaluatestatesavingoverheadandrecoverycostindependently.Wedescribetheana-lyticalmodelswithsettingcheckpointintervalasamajorparameter.Inmanytraditionalmodels,overheadsinnormalandrecoveryoperationsareoftenevaluatedinanintegratedmannerbydefiningthefailurefrequency[Chandyetal.(1975);NicolaandSpanje(1990);Ozakietal.(2004);SolimanandElmaghraby(1998)].However,suchmodelshaddifficultiesinthatthefailurefrequency,whichisverydifficulttospeculateinrunningsystems,hasasignificantimpactontheevaluation.Thischapterisorganizedasfollows.Insection2,weproposeacom-binationaluseofcoordinatedanduncoordinatedcheckpointingwithusingtheincrementalsnapshots.Weconstructanalyticalmodelsevaluatingstatesavingoverheadandrecoverycostoftheproposedtechniqueinsection3andpresentnumericalexamplesanddiscussionsinsection4.Finally,sec-tion5concludethechapter.2ProposedTechnique:HybridCoordinatedCheckpointingUsingIncrementalSnapshotsInthissection,weproposeacheckpointingtechniquewhichcombinesco-ordinatedanduncoordinatedcheckpointingbysubstitutingtheincremen-talsnapshotsfrommostofuncoordinatedcheckpointing.Weusethedis-tributedsnapshotalgorithmproposedbyChandyandLamport[ChandyandLamport(1985)]forcoordinatedcheckpointing.Sinceeachprocessindependentlygeneratestheincrementalsnapshots,weusetheonlineal-gorithmofuncoordinatedcheckpointingintroducedbyJefferson[Jefferson(1985)]forrecovery.Figure1illustratestheconceptoftheproposedcheckpointingtechnique.Eachprocesshasalocalclockmaintainingitsownlocaltime.Thelocalclockisresetto0atthesysteminitiationandeverycompletionofthecoordinatedcheckpointingalgorithm;itisincrementedbyoneforeveryevent.Eventsaregeneratedinternallyoraredeliveredbymessagesfromotherprocesses.Unlikewiththeglobalclockindicatingrealtimesharedinthewholedistributedsystem,theprogressratesofthelocalclocksdiffer 212September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book198ReliabilityModelingwithApplications0T2T㻔S㻙1㻕T0㻼㻝TT2T㻼㻞T2T㻔S㻙3㻕T㻼㻟㻨TCoordinatedcheckpointIncrementalsnapshotFig.1Conceptualdiagramoftheproposedcheckpointingtechniquebetweenprocesses.Foreveryapplicationmessagesentbyaprocess,theprocesssavesitsdestination,themessageidentifier,andthelocaltimeatwhichitissentintothestablestorageinordertoenabletheprocesstogeneratethecorrespondinganti-messageinrecovery.Everytimethelocalclockgoesaconstantnumberofticks(Tticks),theprocessgetachancetosaveitsstateandgenerateanincrementalsnapshot.Theprocesssaveschangesinitsstateandthehistoryofmessagesreceivedafterthelastcoordinatedcheckpointorthelastsnapshot.Thelocalclockisnotadvancedbythesnapshotcreation.Repeatingthesnapshottingcycle,aprocessreachestheSthchanceforsnapshotting,andthen,itinvokesthecoordinatedcheckpointingalgorithmasaninitiatorandsavesitswholestateasacheckpoint.Theremaybetwoormoreinitiatorsincoordinatedcheckpointing.Inaddition,whenanotherprocessreachestheSthchanceduringtheinitiatorsaresavingtheirstates,italsobecomesaninitiatorprocess.Aftercompletionofsavingtheinternalstateinaninitiator,itsendsacontrolmessagetoalloftheotherprocesses.WhenaprocessreceivesthecontrolmessagebeforeitcreateSsnapshots,itgivesuptheremainingsnapshottingchancesandparticipatesthecoordinatedcheckpointing:italsostartssavingofitswholeinternalstate.Therefore,eachprocessinitiatescoordinatedcheckpointingwhenitslocalclockreachesT·Sorwhenitreceivesthecontrolmessage.Aftercompletionofthecoordinatedcheckpointing,alltheoldcheck-points,theincrementalsnapshots,andmessagehistoriesarediscarded.Inaddition,thelocalclocksofallprocessesareresetto0. 213September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookHybridCoordinatedCheckpointingTechniqueUsingIncrementalSnapshots199Hereafter,wedenoteTaslocalcheckpointintervalandSasprescribedcheckpointnumber.WealsodenoteproductofTandS,W≡S·Tascoor-dinatedcheckpointinterval.NotethattheproposedtechniquewithS=1,thatisW=T,isthesameasthetraditionalcoordinatedcheckpointingtechnique.Inthischapter,weexaminetwoincrementalsnapshottechniques:thecumulativesnapshottingtechniqueandthedividedsnapshottingtechnique.Usually,thecumulativetechniquehashigheroverheadforgeneratinganincrementalsnapshot.However,amountoftheoverheaddependsondataupdatepatternofapplications.Forexample,inapplicationswheredataitemsareaddedorupdateduniformlyinwholedataspace,theoverheadofthedividedtechniquecanbealmostconstantwhilethatofthecumulativeonelinearlygrowswithtime.Ontheotherhand,inapplicationshavinghotspots,overheadsofthetwotechniquesaresometimescomparablebecausedataitemsinthehotspotsareintensivelyupdated.Recoveryuponaprocessfailurecanberealizedbyusingtheincrementalsnapshotsinthesimilarmannerasoneillustratedin[Jefferson(1985)].Notethattheincrementalsnapshotsholdonlydifferencesinprocessstates,changedafterthelastcheckpointorsnapshot.Therefore,processeshavetomergethecheckpointdatawiththesnapshotdatainordertorollbacktothestatessavedinthesnapshots.Inthecumulativetechnique,aprocessonlyhastodirectlymergethelastcheckpointdatawiththesnapshot,towhichitwouldreturn.Incontrast,inthedividedtechnique,theprocessmustmergeallsnapshotsgeneratedafterthelastcheckpointrepeatedly.Thus,recoverycostofthedividedoneisusuallyhigherthanthatofthecumulativeone.Althoughthedominoeffectcanoccurinrecoveryoftheproposedtech-nique,arecoverylineisalwaysfoundatacoordinatedcheckpoint.Thus,wecanexplicitlyestimateupperboundofrecoverycost.3EvaluationModelInthissection,weconstructstochasticmodelsforvalidatingeffectivenessoftheproposedtechnique.Inthemodels,coordinatedcheckpointintervalWisusedasoneofmajordesignparameters.First,wedescribeamodelestimatingtheexpectedoverheadforstatesavingintime,H(W),addedtoeveryordinaryeventexecution.Next,weestimaterecoverycostR(W)byderivingtheexpectedtimetakenforrollbackandrerunningafterafailure. 214September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book200ReliabilityModelingwithApplications3.1StateSavingOverheadWedenotetimeintervalbetweentwosuccessivecoordinatedcheckpointingexcludingtimeforgeneratingsnapshotsasX.WefirstderivetheexpectedvalueE[X]forX.SupposethatthereareNprocesses,P1,P2,···,PNinasystem.WedescriberealtimetakenforthelocalclocksreachesWintheprocesses,excludingtimeforgeneratingsnapshots,byX1,X2,···,XN,respectively.Thesearediscreterandomvariables,whichareindependentfromeachother.Definitelyfromtherelationbetweenlogicaltimeandrealtime,W≤Xk(1≤k≤N).Ifweassumeeventsoccurataconstantratepinaunittime,therandomvariablesX1,X2,···,XNobeynegativebinomialdistributions(alsocalledasPascaldistribution).Theprobabilityfunctionisf(x)=Pr{Xk=x}()x−1Wx−W=p(1−p).(1)W−1Notethatx=W,W+1,W+2,···.Also,thecumulativedistributionfunctionis∑xF(x)=f(x′).(2)x′=WBecausethecoordinatedcheckpointingisinitiatedbyaprocesswhichfirstreachestheSthsnapshot,X=min{X,X,···X}.Defining12NPr{X≤W−1}≡0andF(W−1)≡0,theexpectedvalueofXisobtainedby∑∞E[X]=x·Pr{X=x}x=W∑∞=x·[Pr{X≤x}−Pr{X≤x−1}]x=W∑∞([][])NN=x·1−{1−F(x)}−1−{1−F(x−1)}x=W∑∞=W+{1−F(x)}N.(3)x=WNext,weestimateoverheadfortheincrementalsnapshotting.Theoverheadcanbeestimatedbyaccumulatingtimetakenforgenerating 215September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookHybridCoordinatedCheckpointingTechniqueUsingIncrementalSnapshots201S−1(=W/T−1)timessnapshottingintheinitiatorprocess.Wecanassumedispersionoftimeforgeneratinganincrementalsnapshotisneg-ligibleagainstthecoordinatedcheckpointinginterval.Thatis,timeforgeneratingthesthincrementalsnapshotC(s)(1≤s≤S−1)cansnapbedeterministicallyobtained.Concretelyspeaking,weexpresstimeforgeneratinganincrementalsnapshotbythecumulativetechniqueasCsnap(s)=γ+δ(s−1).(4)γdenotestimeforgeneratingthefirstsnapshotandδisincrementaltimeforsavingadded/updateddataitemsinthesecondorlatersnapshotting.Ontheotherhand,timeforthedividedsnapshottingcanbeexpressesasaspecialcasewhereδis0inEq.(4),i.e.,Csnap(s)isalwaysγ.Wenextderiveoverheadforaprocesstoperformcoordinatedcheck-pointing.Today,messageoverheadisverysmallanditcanbenegligiblecomparedtotimeforsavingwholeprocessstates.Moreover,sincetimeforgeneratingacoordinatedcheckpointineachprocessismuchshorterthanthecoordinatedcheckpointinterval,wecanassumethecoordinatedcheckpointingoverheadisconstant.Therefore,wesupposethattimeforcoordinatedcheckpointingcanbeestimatedastimeintervalfromstartofstatesavinginthefirstinitiatorprocesstillcompletionofstatesavingintheprocesswhichislastlyinvolvedinthecheckpointing.WedefineXmin=min{X1,X2,···,XN}andXmax=max{X1,X2,···,XN}forthecumulativerealtimeX1,X2,···,XN.WealsodefineY=Xmax−Xmin.YmeansthetimeintervalfromthelocalclockinthefirstprocessreachesWtotheclockinthelastprocessdoes.WealsodenoteascendingorderedrandomvariablesforthecumulativetimeinNprocessesbyX(1),X(2),···,X(N).Thatis,X(1)≤X(2)≤···≤X(N).Asmentionedearlier,wesupposethesynchronizedalgorithmintroducedbyChandyandLamport[ChandyandLamport(1985)]forcoordinatedcheckpointing.Inthisalgorithm,aninitiatorsendscontrolmessagesafteritsavescheckpointdata,andthecontrolmessagesinvokethealgorithminotherprocesses.Therefore,iftimeneededtosavecheckpointdataineachprocessisC0,thealgorithmisinvokedinallprocesseswithinC0fromitisinvokedinthefirstinitiator.Thus,wecanobtainoverheadfortotalcoordinatedcheckpointingby{Y+C0(Y 216September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book202ReliabilityModelingwithApplicationsderivedasG(y)=Pr{Y≤y}=Pr{Xmax−Xmin≤y}=Pr{Xmax≤Xmin+y}∑∞=Pr{Xmin=x∩Xmax≤x+y}x=W∑∞∑N=Pr{X(1)=X(2)=···=X(k)=x∩x=Wk=1x+1≤X(k+1),X(k+2),···,X(N)≤x+y}∑∞∑N()NkN−k=f(x)·{F(x+y)−F(x)}.(6)kx=Wk=1Thus,expectedoverheadforcoordinatedcheckpointingisC∑0−1E[v(Y)]={1−G(y)}+C0.(7)y=0Combiningoverheadsderivedabove,weobtainstatesavingoverheadperunittimeas∑S−1s=1Csnap(s)+E[v(Y)]H(W)=.(8)E[X]3.2RecoveryCostWhenaprocessrollsbackbyafailure,otherprocessesmayalsoberequiredtorollbackbecauseofanti-messages.Processescanresumeprocessingoflosttasksafterthecascadingrollbacksreacharecoveryline.Notethatsinceeventsnon-deterministicallyoccurinprocesses[Elnozahyetal.(2002)],aprocesswhichrerunsuptothepointatwhichthefailureisdetecteddoesnotnecessarilyresumethesamestatetheprocesshadbeforethefailure.Inthisstudy,weestimaterecoverycostbytimeintervalfromaprocessdetectsafailureuntilitcompletesrerunning.WediscusstheexpectedrecoverycostR(W)alongtherecoveryprocedure.Eachprocessholdsthehistoryofmessagessentafterthelastcoor-dinatedcheckpointinadditiontocheckpoint/snapshotdatainthestablestorage.Thisenablesaprocesstogeneratenecessaryanti-messagesbe-forerollingback.Numberofanti-messagesgeneratedisatmost(N−1) 217September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookHybridCoordinatedCheckpointingTechniqueUsingIncrementalSnapshots203forNprocessesinthesystem[MatsumotoandTaki(1996);Elnozahyetal.(2002)].Supposingeachentryinthemessagehistoryisverysmallandrandom-accessible,wecanregardtimeneededtogenerateanti-messagesnegligiblecomparedtothatforreadingcheckpointdatafromthestablestorage.Similarly,timeforsendinganti-messagesandmessagedelayarealsosmall,thereforeweassumethatrecoveryoperationsareinvokedatthesametimeinallprocessesinvolved.TimeintervalbetweentwosuccessivecoordinatedcheckpointscanbedividedintoSlocalintervalsby(S−1)incrementalsnapshots.Thelocalintervalsrepresentfortimepointsoffailures.Ifaprocessfailsinthesthlocalinterval(0≤s≤S−1),itholdssincrementalsnapshotsatthebeginningofrecovery.WedenotetimecostfortheprocessreconstructingitsstatesavedatthelateststhincrementalsnapshotbyC(s).Regardingloadthelastcoordinatedcheckpointasthe0thincrementalsnapshot,Cload(s)=C0+Csnap(s)(9)forthecumulativesnapshottingand∑sCload(s)=C0+Csnap(i)(10)i=1∑0forthedividedsnapshottingwherewedefineCsnap(0)≡0andi=1·≡0.Wealsoassumethattimeforreadingcheckpoint/snapshotdataequalstotimeforgeneratingthem.Timecostneededforthecascadingrollbackreachesarecoverylineandtimeforrerunningthelosttasksdependontheprocesswhichhasthegreatestrollbackscaleinthesystem;wedenotethemaximumnumberofrollbacksinaprocessinarecoverybyZ.Weassumeeachprocessrollsbackonebyoneforeachincrementalsnapshotitholds.TimeintervalfromtheprocesswiththemaximumrollbackscaleZreturnstothestateofthelatestincrementalsnapshottillitreachesarecoverylinecanbeestimatedassumofcostsforreadingtheremainingZ−1snapshots,Csnap(s−1),Csnap(s−2),···,Csnap(s−Z+1).Ifweassumethatfailuresoccuratthemiddlepointofthelocalintervalsinaverage,logicaltimetakenforthefailedprocesstorerunthelosttaskscanbeestimatedasZT−(1/2)T.Thus,theexpectedrealtimeis(Z−1/2)T/p.Inaddition,aprocesshastimecostofCsnap(s−Z+2),Csnap(s−Z+3),···,Csnap(s−1),Csnap(s)inthererunninginordertoregenerate(Z−1)snapshots.Notethattheregenerationcostis0whenZ=1. 218September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book204ReliabilityModelingwithApplicationsTherefore,whentheprocesswiththemaximumZ-timerollbackhassincrementalsnapshotsuponafailure,totalrecoverycostrs(Z)isobtainedbyZ∑−1(Z−1/2)TZ∑−1rs(Z)=Cload(s)+Csnap(s−i)++Csnap(s−i+1),pi=1i=1(11)whererandomvariableZobeysaprobabilitydistributionbs(z),thatis,bs(z)=Pr{Z=z}.IfweassumeprocessfailurescanoccuruniformlyintheSlocalintervals,theexpectedrecoverycostR(W)isgivenbyS∑−11∑s+1R(W)=bs(z)·rs(z).(12)Ss=0z=1NotethatS=W/T.Theprobabilitybs(z)dependsonthelocalandcoordinatedcheckpointintervals,numberofprocesses,messagefrequency,communicationpatterns,andsoon.Itisquitedifficulttoanalyticallyderivebs(z)exceptinspecialcasewherethereareafewprocesses.Inthisstudy,weevaluatethedistri-butionthroughMonteCarlosimulations.Inthesimulations,werandomlygenerateeventsofmessageexchangeandinternalprocessingandadvancethelocalclocks.Messageeventsaregeneratedatprobabilityq.4NumericalExamplesandDiscussionsInthissectionwepresentnumericalexamplessuggestingtrade-offrelationsbetweenstatesavingoverheadH(W)andrecoverycostR(W)basedontheproposedanalyticalmodels.Throughthenumericalexamples,wediscusseffectivenessoftheproposedanalyticalmodels.Figure2showsthestatesavingoverheadforthecumulativesnapshot-ting,thedividedsnapshotting,andatraditionalcoordinatedtechnique(thealgorithmproposedbyChandyandLamport).Thetimeforgeneratingacheckpointincoordinatedcheckpointing,C0,wassetto25,andtheparame-terfortheincrementalsnapshottingγwas5,respectively.Inthecumulativesnapshotting,wesetδ=2.WefixedT=100andchangedtheprescribedcheckpointnumberSsothatW=100Sintheproposedtechnique.Be-causeSisalways1inthetraditionaltechnique,TdefinitelyequalstoW.Allexamplespresentedinthesectionusethesameparameters. 219September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookHybridCoordinatedCheckpointingTechniqueUsingIncrementalSnapshots2050.50.45cumulative(N=64)divided(N=64)0.4traditional(N=64)0.35cumulative(N=1024))0.3divided(N=1024)traditional(N=1024)(W0.25H0.20.150.10.0501002003004005006007008009001000WFig.2StatesavingoverheadH(W)oftheproposedandatraditionalcoordinatedcheckpointingtechniques(p=0.9,C0=25,γ=5,andδ=2)Inthetraditionaltechnique,H(W)monotonicallydecreaseswithWbecauseoverheadforcoordinatedcheckpointingimposedperunittimeislowerwithlargerW.Intheproposedtechniques,theincrementalsnapshotiscombinedwithcheckpointing.Inthedividedsnapshotting,H(W)alsomonotonicallydecreasesbecauseCsnap(s)isalwaysγandoverheadforgen-eratingsnapshotsisalsoconstant.Ontheotherhand,inthecumulativetechnique,overheadforgeneratingsnapshotsmonotonicallyincreasesbe-causeCsnap(s)increaseswiths(1≤s≤W/T−1).ThisleadsthatH(W)hasthelocalminimumvalueinthecumulativetechnique.InFig.2,thenumberofprocessesNis64or1024.ThelargerNis,thegreaterexpectedtimeforcoordinatedcheckpointingE[v(Y)].Thus,H(W)isgreaterwithlargerN.However,aswecanseeinEq.(5),E[v(Y)]islimitedsothatitisatmost2C0,therefore,H(W)withN=1024isatmost1.2timesthatwithN=64.Next,wediscusstheprobabilitydistributionofthemaximumrollbackdistanceZobtainedthroughsimulations.Whenmessageprobabilityisrelativelylow,i.e.,q=0.01,bs(1)isalmost1foranycombinationsof 220September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book206ReliabilityModelingwithApplications10.9s=1q=0.05s=20.8s=3s=40.7s=5s=60.6s=7s=80.5s=9Probability0.4s=100.30.20.1012345678910MaximumrollbackdistanceFig.3ProbabilitydistributionofthemaximumrollbackdistancewithN=64andq=0.05(S=10,T=100,p=0.9,C0=25,γ=5,andδ=2)N,S,T,C0,γ,δandtimelocationoffailurepoints.Thismeansarecoverylineconsistingofthecurrentstateorthelatestsnapshotofeachprocesscanbeobtainedinalmostallcases.Withhighermessageprobabilityq=0.05,wecanobtaintheprobabilitydistributionasshowninFig.3.ThefigureshowsnumericalexampleswiththesameparameterstoFig.2,i.e.,N=64,C0=25,andCsnap(s)=5+2s.WefixedthecoordinatedcheckpointintervalS=10,T=100,i.e.,W=1000.Eachlinedenotesarollbackdistancedistributionwhenafailureoccursinaprocesswhichholdsssnapshots.Theprobabilitybs(1)isthelargestregardlessofthefailurepoints.Inmanycases,severaltimesofrollbackoccurredduetothedominoeffect,however,probabilityofthataprocessrollsbackuptothecoordinatedcheckpoint,bs(s),isrelativelylow.Therefore,wecanexpectbetterrecoveryperformancefortheproposedtechniquewhencostsformergingtheincrementalsnapshotsaresmallerthanthatforreadingcheckpointdata.Figure4showsaprobabilisticdistributionsformuchhighermessageprobabilityq=0.1.Valuesofparametersotherthanqarethesameas 221September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookHybridCoordinatedCheckpointingTechniqueUsingIncrementalSnapshots2071s=10.9s=20.8q=0.1s=3s=40.7s=5s=60.6s=7s=80.5s=9Probabilitys=100.40.30.20.1012345678910MaximumrollbackdistanceFig.4ProbabilitydistributionofthemaximumrollbackdistancewithN=64andq=0.1(S=10,T=100,p=0.9,C0=25,γ=5,andδ=2)thoseinFig.3.Inthisexample,bs(s),whichistheprobabilitywhereaprocessrollsbackuptothecoordinatedcheckpoint,isthehighest.Insuchsituations,theincrementalsnapshotscannotwellcontributetoreductionofrecoverycost,theyrathermayincreasetherecoverycostoftheproposedtechniquecomparedtothetraditionaltechnique.WeperformedsimilarsimulationswithN=1024andfoundthatgrowthofnumberofprocessesaveragelyincreasedthemaximumrollbackdistanceZ.Wecanreadfromtheseresultsthatthelargermessageprobabilitypisandthemoreprocessesasystemhas,thelargerthemaximumrollbackdistanceZis.WecalculatedrecoverycostR(W)fromEq.(12)andbs(s)obtainedbythesimulationdescribedabove.Figure5showsnumericalexamplesofR(W)formessageprobabilityq=0.05.Otherparametersweresetasp=0.9,C0=25,andγ=5,respectively.Theincrementincostforsnap-shotting,δ,was2forthecumulativetechniqueandalways0inthedividedtechnique.WefixthelocalcheckpointintervalT=100,thatis,W=100SinthetwoproposedsnapshottingtechniquesandT=Winthetraditional 222September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book208ReliabilityModelingwithApplications600cumulative(N=1024)cumulative(N=64)500divided(N=1024)divided(N=64)400traditional)(W300R20010001002003004005006007008009001000WFig.5RecoverycostR(W)oftheproposedandatraditionalcoordinatedcheckpointingtechniques(p=0.9,q=0.05,C0=25,γ=5,andδ=2)technique.R(W)monotonicallyincreasedwithincreaseofW.Inthedi-videdtechnique(δ=0),R(W)increasedwithWalmostlinearly;thisisthesimilartrendtothatofthetraditionaltechnique.Thisisbecauseinthedividedtechniquewehavetorepeatedlymergeallincrementalsnapshotsfromtheonenexttothecoordinatedcheckpointuptotheonetowhichaprocesswouldliketorollback.Ontheotherhand,recoverycostincreasedataslowerpaceinthecumulativetechnique;itisalwayslowerthanthatofthetraditionalone.Weshoulddecidecheckpointing/recoverypolicybasedonrequirementsbothinperformanceandreliabilityofthesystem,consideringtrade-offbetweenthestatesavingoverheadandtherecoverycost.Wenowdiscussvalidityoftheproposedtechnique,gatheringdiscussionsabove.Figures6–8showtrade-offrelationsbetweenstatesavingoverheadH(W)andrecoverycostR(W)withsettingmessageprobabilityq=0.01,0.05or0.1.Inallfigures,numberofprocessesN=64.SimilarlytoFigs.2and5,wesettheparametersasp=0.9,C0=25,γ=5,andδ=2.ThelocalcheckpointintervalTwassetto100andW=100Sintheproposedtechniques;T=Winthetraditionalcoordinatedtechnique. 223September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookHybridCoordinatedCheckpointingTechniqueUsingIncrementalSnapshots209600W=1000W=900q=0.01500W=800W=700400W=600)cumulative(W300W=500dividedRW=400traditional200W=300W=200100W=10001000500300W=200W=500W=400W=300W=200W=10040000.050.10.150.20.250.30.350.4H(W)Fig.6StatesavingoverheadH(W)andrecoverycostR(W)withN=64andq=0.01(p=0.9,C0=25,γ=5,andδ=2)Figure6showsnumericalexamplesforq=0.01.Withsuchalowmes-sageprobability,rollbacksusuallystopatthelatestsnapshotsorcheck-points.Therefore,therecoverycostoftheproposedtechniquewasnotaffectedbyWbecausetheexpectedtimeforrerunningthelosttasksisT/2pregardlessofW.Comparingtothetraditionaltechniqueinwhichthecoordinatedcheckpointintervalhassubstantialimpactontherecov-erycost,theproposedtechniquescanachievefasterrecoverywithgreatlylowerstatesavingoverhead.Uncoordinatedcheckpointingcanalsoreducetheexpectedrecoverycostwhenmessagefrequencyislow,however,thereisnoupperboundoftherecoverycostbecausetheworst-caserollbackdis-tancereachesinfinity[Agbariaetal.(2001)].Theproposedtechniquesarebetterforsystemshavingrestrictionsontherecoverycost.Especially,thedividedtechniquecanreducethestatesavingoverheadwithoutincreasingtherecoverycost.Thereforeitissuitableforapplicationswheremessagesareinfrequentlyexchanged.Figure7presentnumericalexamplesformessageprobabilityq=0.05.Theproposedtechniqueshavelowerrecoverycostthanthatofthetradi- 224September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book210ReliabilityModelingwithApplications600W=1000W=900W=1000500q=0.05W=800W=700400W=600cumulative)(W300W=500dividedRW=500traditionalW=1000W=400200W=400W=300W=500W=300W=200W=400W=200100W=300W=200W=10000.050.10.150.20.250.30.350.4H(W)Fig.7StatesavingoverheadH(W)andrecoverycostR(W)withN=64andq=0.05(p=0.9,C0=25,γ=5,andδ=2)tionaltechniquewhenwecanhavemorethan0.15forstatesavingoverheadH(W)inthecumulativetechniqueand0.1inthedividedtechnique,respec-tively.Moreover,undersituationswithahardrestrictionintherecoverycost,forexampleR(W)≤200,theproposedtechniquescanfulfillthere-strictionwithlowerH(W).Forapplicationswithstrongerrestrictionintherecoverycost,thecumulativetechniqueisveryeffective.Inthecumulativetechnique,sincethereiscoordinatedcheckpointintervalW∗correspondingtothelocalminimumvalueofstatesavingoverhead(W∗=500inFig.7),lettingWmorethanW∗isuselessinpractice.TherecoverycostisalsoboundatmostR(W∗).However,withstrongrestrictionsinstatesavingoverhead,wehavetobalanceH(W)andR(W)byusingthetraditionaltechnique.InFig.8,wesetqto0.1.Withsuchahighmessageprobability,rollbackstothecoordinatedcheckpointfrequentlyoccurduetothedominoeffect,therefore,wecannotexpectfortheproposedtechniquetoimprovetherecoveryperformance.Insuchsituations,costformergingtheincrementalsnapshotsmayratherhaveanegativeimpactonrecoverycost.Thus,the 225September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookHybridCoordinatedCheckpointingTechniqueUsingIncrementalSnapshots211700W=1000600W=1000q=0.1W=900500W=1000W=800W=700)400cumulative(WW=600RW=500divided300W=500W=400W=400W=500traditionalW=400200W=300W=300W=200100W=200W=10000.050.10.150.20.250.30.350.4H(W)Fig.8StatesavingoverheadH(W)andrecoverycostR(W)withN=64andq=0.1(p=0.9,C0=25,γ=5,andδ=2)recoverycostoftheproposedtechniquesislargerthanthatofthetraditionaloneinmanycases.WecanobtainlowerrecoverycostthanthatofthetraditionalonewhenwesetH(W)morethan0.17,however,advantageoftheproposedtechniqueisrelativelysmall.WealsoevaluatetheproposedtechniqueforN=1024.Similarly,theproposedtechniquecanachievebettertrade-offwithsmallmessageprob-abilityqandtheyweaktheiradvantageswithgrowthofq.Moreover,theexpectedrollbackdistanceislargerforN=1024asmentionedabove,therefore,theproposedtechniquelosestheiredgewithsmallerqcomparedtonumericalexamplesforN=64.5ConcludingRemarksInthischapter,weproposedacoordinatedC/Rtechniqueusingtheincre-mentalsnapshotandevaluateditsstatesavingoverheadandrecoverycostbyconstructingsimplestochasticmodels.Inevaluationoftherecoverycost,wespecifiedtheprobabilisticdistributionofthemaximumrollbackdistancethroughsimulations. 226September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book212ReliabilityModelingwithApplicationsWecomparedthestatesavingoverheadH(W)andtherecoverycostR(W)oftheproposedandthetraditionalcoordinatedC/Rtechniques.Usingtheproposedtechnique,wecanachievelowerR(W)withcomparablelowH(W)tothetraditionalone.Eveninenvironmentswithrelativelyhighmessagefrequency,theproposedtechniquescanachievefasterrecoveryifwecanallowsomegrowthinthestatesavingoverhead.Therearedifferentapplicationssuitableforthetwoproposedsnapshot-tingtechniques,thedividedtechniqueandthecumulativetechnique.Thedividedtechnique,whichhasnolowerboundinH(W),isbetterforen-vironmentswherethemessagefrequencyislowenoughandthedominoeffectsseldomoccur.Thecumulativetechniqueissuitableforapplicationsinwhichmessagesareexchangedmorefrequently.TheproposedtechniqueutilizesChandyandLamport’scheckpointingalgorithmwhereallprocessesinasystemsynchronouslysavetheirstate.ItispossiblethatusingmoreefficientcoordinatedalgorithmssuchasoneproposedbyKooandToueg[KooandToueg(1987)]improvesperformanceandreliabilityofdistributedsystems.Wewilltackledevelopingsuchmoreefficienttechniques.ReferencesAgbaria,A.,Attiya,H.,Friedman,R.andVitenberg,R.(2001).QuantifyingthRollbackPropagationinDistributedCheckpointing,Proc.20IEEESympo.ReliableDistributedSystems(SRDS’01),pp.36–45.Camargo,R.,Goldchleger,A.,Kon,F.andGoldman,A.(2004).Checkpointing-BasedRollbackRecoveryforParallelApplicationsontheInteGradeGridndMiddleware,Proc.2WorkshoponMiddlewareforGridComputing,pp.35–40.Chandy,K.M.,Brown,J.C.,Dissly,C.W.andUhrig,W.R.(1975).AnalyticModelsforRollbackandRecoveryStrategiesinDataBaseSystems,IEEETrans.Softw.Eng.,Vol.SE-1,pp.100–110.Chandy,K.M.,Lamport,L.(1985).DistributedSnapshots:DeterminingGlobalStatesofDistributedSystems,ACMTrans.Comput.Syst.,Vol.3,No.1,pp.63–75.Elnozahy,E.N.andZwaenepoel,W.(1992).Manetho:TransparentRollback-RecoverywitLowOverhead,LimitedRollbackandFastOutputCommit,IEEETrans.Comput.,Vol.41,No.5,pp.526–531.Elnozahy,E.N.,Alvisi,L.,Wang,Y.-M.andJohntson,D.B.(2002).ASurveyofRollback-RecoveryProtocolsinMessage-PassingSystems,ACMComput.Surv.,Vol.34,No.3,pp.375–408.Koo,R.andToueg,S.(1987).CheckpointingandRollback-RecoveryforDis-tributedSystems,IEEETrans.Softw.Eng.,Vol.SE-13,No.1,pp.23–31. 227September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookHybridCoordinatedCheckpointingTechniqueUsingIncrementalSnapshots213Jefferson,D.(1985).VirtualTime,ACMTrans.Prog.Lang.Syst.,Vol.7,No.3,pp.404–425.Krishnan,S.andGannon,D.(2004).CheckpointandRestartforDistributedCom-thponetnsinXCAT3,Proc.5IEEE/ACMInt’lWorkshoponGridCom-puting(GRID’04),pp.281–288.Matsumoto,Y.andTaki,K.(1996).EfficientTechniqueforRealizingTimeWarpMechanismforParallelLogicSimulation,Trans.IPSJ,Vol.37,No.4,pp.654–665(inJapanese).Nicola,V.F.andVanSpanje,J.M.(1990).ComparativeAnalysisofDifferentModelsofCheckpointingandRecovery,IEEETrans.Softw.Eng.,Vol.SE-16,No.8,pp.807–821.Ozaki,T.,Dohi,T.,Okamura,H.andKaio,N.(2004).Min-MaxCheckpointPlacementunderIncompleteFailureInformation,Proc.IEEEInt’lConf.DependableSystemsandNetworks(DSN2004),pp.721–730.Soliman,H.M.andElmaghraby,A.S.(1998).AnAnalyticalModelforHybridCheckpointinginTimeWarpDistributedSimulation,IEEETrans.ParallelDistrib.Syst.,Vol.9,No.10,pp.947–951.Tohma,Y.andMukaidono,M.ed.(1988).IntroductiontoReliabilityTechniquesforComputerSystems,(JapanStandardsAssociation)(inJapanese).Woo,N.,Jung,H.,Yeom,H.,Park,T.andPark,H.(2004).MPICH-GF:Trans-parentCheckpointingandRollback-RecoveryforGrid-EnabledMPIPro-cesses,IEICETrans.Inf.&Syst.,Vol.E87-D,No.7,pp.1820–1828. 228May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank 229September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookChapter12PredictingEffortandErrorsforEmbeddedSoftwareDevelopmentProjectsbyUsinganArtificialNeuralNetworkKazunoriIwata1,SayoriMaeji2andToshioNakagawa31DepartmentofBusinessAdministration,AichiUniversity,4-60-6Hiraike-cho,Nakamura,Nagoya453-8777,Japan2InstituteforConsumerScienceandHumanLife,KinjoGakuinUniversity,1723Omori2,Moriyama,Nagoya463-8521,Japan3DepartmentofBusinessAdministration,AichiInstituteofTechnology,1247Yachigusa,Yakusa-cho,Toyota470-0392,Japan1IntroductionRecently,growthintheinformationindustryhascausedawiderangeofusesforinformationdevices,andtheassociatedneedformorecomplexembed-dedsoftware,thatprovidesthesedeviceswiththelatestperformanceandfunctionenhancements[Hirayama(2004);Nakamotoetal.(1997)].Con-sequently,itisincreasinglyimportantforembeddedsoftware-developmentcorporationstoascertainhowtodevelopsoftwareefficiently,whilstguaran-teeingdeliverytimeandquality,andkeepingdevelopmentlowcosts[Boehm(1976);Watanabe(2004);Tamaru(2004)].Hence,companiesanddivisionsinvolvedinthedevelopmentofsuchsoftwarearefocusingonvarioustypesofimprovement,particularlyprocessimprovement.Predictingeffortrequire-mentsofnewprojectsandguaranteeingqualityofsoftwareareespeciallyimportant,becausethepredictionrelatesdirectlytocosts,whilethequal-ityreflectsonthereliabilityofthecorporation[OgasawaraandKojima215 230September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book216ReliabilityModelingwithApplications(2003);Komiyama(2003);Takagi(2003);Nakashima(2004);N.(2004)].Inthefieldofembeddedsoftware,developmenttechniques,managementtechniques,tools,testingtechniques,reusetechniques,real-timeoperatingsystemsandsoon,havealreadybeenstudied.However,thereislittlere-searchontherelationshipbetweenthescaleofthedevelopmentandthenumberoferrors,basedofdataaccumulatedfrompastprojects.Asaresult,previouslywedescribedthepredictionofthetotalscaleusingmul-tipleregressionanalysis[Iwataetal.(2006);Nakashimaetal.(2006)].Inthischapterwetherefore,proposeamethodforcreatingeffortanderrorspredictionmodelusinganArtificialNeuralNetwork(ANN).However,theANNhasalargemarginoferrorsforsomeprojects.Wetherefore,proposeamethodtoreducethemarginoferrorsmodel.Finally,wealsocomparetheaccuracyoftheproposedANNmodelwiththatofamultipleregressionanalysismodelusingWelch’st-test[Student(1908);Welch(1947)].Therestofthechapterisorganizedasfollows.InSection2,weex-plainsoftwaredevelopmentmanagementanddiscusscurrentproblemsandtheobjectiveswhichthisstudyistryingtoachieve,andillustratesoftwaredevelopmentprocessandselectionofdatatoestablishthemodelinSec-tion3.InSection4weexpoundmodelstopredicteffortanderrors.InSection5describesevaluationexperiment.Section6concludes.2SoftwareProjectManagementandIssuesTheembeddedsoftwareforfinancialinstitutionsdevelopedby“OMRONsoftwareCo.”isbasedonthebasicsoftwarecustomizedforanindividualcustomer’sneedtoinstallitatvarioussites.Tominimizethecustomizationneededduringthedevelopmentofbasicsoftware,parametersareembed-dedtocontrolthesystem.Thisengineeringtechniqueassuresproductivityandquality.Thistypeofapproachisactivelytakenduringtheprocessofsoftwaredevelopment.Whileusingthistypeoftechnique,thepressuresrelatedtodeliverytimeandqualityaremoreandmoreintense.Thisre-quiresimprovingfurtherthequalityandcostduringsoftwaredevelopment.Hence,wehavealreadystudiedcostsoftheprocessesbyusinganalysis[Iwataetal.(2006);Nakashimaetal.(2006)].Tocopewiththissituation,thetoolsthatcanmanagetheprogressstatusorresultsinthedatabaseareusedtoimprovethequalityandproductivity.However,themorethevolumeofsoftwaredevelopmentprojectincreasesthemoretheerrorsin-crease.Byanalyzingthedatabase,wedeterminetherelationshipbetweenthevolumeofsoftwaredevelopmentprojectandtheerrors. 231September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookPredictingEffortandErrorsbyUsinganANN2173SoftwareDevelopmentProcessesandSelectionofDataInsoftwaredevelopmentdivisionof“OMRONsoftwareCo.”,thewaterfallmodel[Boehm(1976)]isusedasthebasicdevelopment-processmodel.AgeneraldescriptionofthismodelisgiveninTable1.Table1SoftwareDevelopmentProcess.ProcessContentsofwork1Conceptualdesign(CD)Thisisso-called“systemengineeringwork”.Theyan-alyzecustomerrequirementsanddetailtheareastobeaddressedasdevelopmentfactors.2DesignAccordingtothedevelopmentfactorsdefinedinCDprocess,designingofsoftwarefunctionality,combiningofsoftwaremodules,andwritingofsourcecodeareperformed.3DebuggingVerifytheoutcomeofthedesignprocesswiththeac-tualmachinetoseeifitisdesignedaccordingtothedesign.Thesamedesignerindesignprocessisassignedtodebug.4TestAfterfinishingdebugging,double-checkthesoftwaretoconfirmthatitsatisfiescustomer’srequirements.Adifferentperson(notthoseassignedtodesignanddebug)isassignedtodothis.Thesoftwaredevelopmentofeachindividualpieceofequipmentiscalled“aproject”,anddefinedasfollows.Definition12.1.ProjectThesoftwaredevelopmentforeachpieceofequipmentthatconstitutesthesystemiscalled“thedevelopmentproject”.Eachdevelopmentprojectneedscostofhuman,whichiscalled“effort”andmakessomefaults,whichiscalled“errors”definedasfollows.Definition12.2.EffortMan-dayscostforsoftwaredevelopmentprojects.Definition12.3.ErrorsFaultsinamachine,especially,inasystemorprogram.Regardingthedatarelatedtoprojectproductivityorquality,dataforsuchthingsasinternaleffortandsizeinformationetc.arerecordedasshowninTable2andTable3,for3pointsintime: 232September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book218ReliabilityModelingwithApplicationsTable2ClassificationsandSelectionofData1.ItemsDataSelectionReasonInternalef-EffortSelectedThedataareacquiredbyeffortfortforeachplannedmanagementsystem.Theseprocessandactualdataarequantitativeandac-performancecuracyisgood.Project-NumberoflinesSelectedThesedataarequantitativescaleinfor-innew,modified,andaccuracyisgood.mationoriginalandreusedsoftwareEffortin-ActualeffortinRejectedThedataareacquiredattheformationeachprocessendoftheproject.Hence,theycannotusetopredicteffortanderrorsatthebeginningoftheproject.EffortforredoinRejectedThedefinitionofredoisnoteachprocessconsistent.Theaccuracyispoor.1)Atthebeginningoftheproject.2)Duringtheproject.3)Attheendoftheproject.Beforeanalyzingdata,weexaminedthedataanddecidedwhichdatashouldbeselectedtomakethemodel.Table2andTable3showthelatterdataandtheresultsoftheselection.3.1DataSetsforCreatingModelsUsingthefollowingdata,wecreatemodelstopredictboththeplanningeffort(Eff)anderrors(Err).Eff:“Theamountofeffort”,whichneedsbepredicted.Err:“Thenumberoferrors”inaproject.Vnew:“Volumeofnewlyadded”,whichdenotesthenumberofstepsinthenewlygeneratedfunctionsofthetargetproject.Vmodify:“Volumeofmodification”,whichdenotesthenumberofstepsmodifyingandaddingtoexistingfunctionstousethetargetproject.Vsurvey:“Volumeoforiginalproject”,whichdenotestheoriginalnumberofstepsinthemodifiedfunctions,andthenumberofstepsdeletedfromthefunctions.Vreuse:“Volumeofreuse”,whichdenotesthenumberofstepsinfunctionsofwhichonlyanexternalmethodofhasbeenconfirmedandwhich 233September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookPredictingEffortandErrorsbyUsinganANN219Table3ClassificationsandSelectionofData2.ItemsDataSelectionReasonProductsProductclassifica-SelectedItisnecessarytomakechar-informa-tionandproductacteristicsoftheproductsandtionmodelsdevelopmentprocessbere-flectedinthemodel.CustomernameRejectedThedataisqualitativeanditandsubprojectisdifficulttoobtainaccuratenamedata.DevelopmentRejectedBecausethereareonlytwotype(newormodi-types,itisnotappropriateasfication)parameterforthemodel.DeliverytimeRejectedBecausethedeliverytimeisseldomchanged,thisisnotse-lected.OutsourcingTheestima-RejectedBecauseoutsourcingamounttionofoutsourcingincludessalesaspects,thisamountandactualisnotappropriateforactualsituationprojecterrorstatus.TheestimationofRejectedBecausethisisestimatedbyoutsourcingeffortoutsourcingamountandin-andactualsitua-cludessalesaspects,thisisnottionselected.Qualityin-Thenum-SelectedItisnecessarytofindrelation-formationberofproblemsinshipbetweenaprojectander-eachprocessrors.Thesedataconsistof“TotalError”,“ErrorinCDandDesign”,“ErrorinDebug-ging”and“ErrorinTest”.areappliedtothetargetprojectdesignwithoutconfirmingtheinternalcontents.4EffortandErrorPredictionModels4.1AnArtificialNeuralNetworkModelArtificialNeuralNetworks(ANNs)areessentiallysimplemathematicalmodelsdefiningfunction.f:X→Y(1)whereX={xi|0≤xi≤1,i≥1}andY={yi|0≤yi≤1,i≥1}.ANNsarenon-linearstatisticaldatamodelingtoolsandthatcanbeusedtomodelcomplexrelationshipsbetweeninputsandoutputs.The 234September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book220ReliabilityModelingwithApplicationsOutputw’1w’2HiddenNode1HiddenNode2w1,1w1,2w2,1w2,2Input1Input2Fig.1BasicArtificialNeuralNetworkbasicmodelisillustratedinFig.1,inwhichtheoutputiscalculatedasfollows.1)Calculatingvaluesforhiddennodes.ThevalueofHiddenNodejiscalculatedusingthefollowingequation:()∑HiddenNodej=f(wi,j×Inputi)(2)iwheref(x)equals1andthewisweightcalculatedbythe1+exp(−x)i,jlearningalgorithm.2)CalculatingOutputusingHiddenNodejasfollows:()∑Output=f(w′×HiddenNode)(3)kkkwheref(x)equals1andthew′isweightcalculatedbythe1+exp(−x)klearningalgorithm.WecanuseanANNtocreateeffortanderrorpredictionmodels.4.1.1NormalizationofdataInanANN,arangeofinputvaluesoroutputvaluesisusuallylessthanorequalto1andgreaterthanorequalto0.However,mostselecteddataaregraterthan1.Eachdatarangeis,therefore,convertedto[0,1]bynormalization.Thenormalizedvaluefortkindisexpressedasfn(tkind) 235September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookPredictingEffortandErrorsbyUsinganANN221(wherekinddenotesEff,Err,Vnew,Vmodify,VsurveyandVreuse).Thenormalizedvaluefn(tkind)iscalculatedusingEq.(4).tkind−min(Tkind)fn(tkind)=(4)max(Tkind)−min(Tkind)whereTkinddenotesthesetoftkind,andmax(Tkind)andmin(Tkind)denotethemaximumandminimumvalues,respectively,ofTkind.Thenormalizationisflatandsmooth,then,asmallchangeinanormal-izedvalueinfluencesasmall-scaleprojecttoagreaterdegreethanalargescaleproject.Forexample,letmin(TEff)equal10,max(TEff)equal300,tEff1equal15,tEff2equal250,predictedvaluefortEff1bebtEff1andtEff2bebtEff2.Ifthepredictionmodelhas+0.01error,thenf−1(0.01)=2.90.Thepre-nldictedvaluesresultinbtEff1=17.90andbtEff2=252.90.Bothcaseshassameerrors,buttheirabsoluteoftherelativeerrors(ARE)arefollows:bt−t17.90−15ARE=Eff1Eff1==0.1933Eff1t15Eff1bt−t252.90−250ARE=Eff2Eff2==0.0116Eff2t250Eff2Theresultsindicatetheabsoluteoftherelativeerrorsofformerisgreaterthanthatofthelatter.Thesedistributionsfortheamountofeffortandthenumberoferrorsindicatethesmall-scaleprojectsaremajorandmorethanthelargescaleprojects.Therefore,inordertoimprovepredictionaccuracy,itisimportanttoreconstructthenormalizingway.4.2NewNormalizationofDataInordertosolvetheproblem,weadoptnewnormalizingwayinthefollow-ingequation:√fn(t)=1−(fn(t)−1)2(5)clThecomparisonbetweenEq.(4)andEq.(5)isshowninFigs.2and3.TheEq.(5)hasasharpinclinationattheloweroriginaldata,thenasmallchangeattheloweroriginaldatagetmagnified.Usingthesameassumption,thepredictedvaluesresultinbtEff1=15.56andbtEff2=271.11.TheirabsoluteoftherelativeerrorsareinEq.(6)and 236December19,201311:42BC:9023-ReliabilityModelingwithApplications2013book222ReliabilityModelingwithApplicationsFig.2NormalizingResultsusingEq.(4)Fig.3NormalizingResultsusingEq.(5)Eq.(7).15.56−15AREEff1==0.0373(6)15271.11−250AREEff2==0.0844(7)250Theresultsshowtheabsoluteoftherelativeerrorsforthesmall-scaleprojectissmallerthanthatofoldnormalizationmethodand,incontrast, 237September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookPredictingEffortandErrorsbyUsinganANN223!"#$"#%&'()*+#&*+&(++*+%,!-./0!-1*2345!-%"+6/5!-+/./0!Fig.4StructureofModelthatforthelargescaleprojectisslightlylargerthanthatofoldnormaliza-tionmethod.ThemoredetailedcomparisonanalysesareinSection5.4.2.1StructureofmodelInafeed-forwardANN,theinformationismovedfrominputnodes,throughthehiddennodestotheoutputnodes.Thenumberofhiddennodesisim-portant,becauseifthenumberistoolarge,thenetworkwillover-training.Thenumberofhiddennodesis,generally2/3ofthenumberofinputnodesortwicethenumberofinputnodes.Inthischapter,weuse8hiddennodesinourmodelwhichisillustratedinFig.4.4.3MultipleRegressionAnalysisModelThemultipleregressionanalysis(MRA)modelisderivedfromEq.(8),inwhichthenotationadherestothemeaningsdefinedinSubsection3.1.ThemodelselectsexplanatoryvariablesbyusingabestsubsetselectionprocedurebasedonAkaike’sInformationCriterion(AIC)[Akaike(1973)].ThesmallertheAICvalue,thehighertheaccuracyorfitofthemodel.D=α×S2+α×S+β(8)12 238September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book224ReliabilityModelingwithApplicationswhere,DindicatesEfforErr,andSiscalculatedbyEq.(9).S=Vnew+Vmodify+θ1×Vsurvey+θ2×Vreuse(9)where,θ1andθ2arelessthan1,thusEq.(9)emphasizesVnewandVmodify.5EvaluationExperiment5.1EvaluationCriteriaEquations(10)to(13)areusedasevaluationcriteriafortheeffortanderrorspredictionmodels.Thesmallerthevalueofeachevaluationcriterion,thehigheristherelativeaccuracyinEqs.(10)to(13).TheaccuracyvalueisexpressedasX,andthepredictedvalueasXb.Also,thenumberofdataisexpressedasn.1)MeanofAbsoluteErrors(MAE).2)StandardDeviationofAbsoluteErrors(SDAE).3)MeanofRelativeErrors(MRE).4)StandardDeviationofRelativeErrors(SDRE).1∑MAE=|Xb−X|(10)n√1∑()2SDAE=|Xb−X|−MAE(11)n−11∑Xb−XMRE=(12)nXvu()2u∑Xb−Xt1SDRE=−MRE(13)n−1X5.2DataUsedinEvaluationExperimentTheevaluationexperimentusesthedataforrealprojects.Theprojectdataisdividedintotworandomsets.Oneofthetwosetsisusedastrainingdata,whiletheotheristestdata.Thetrainingdataisusedthegenerationoftheeffort(orerrors)predictionmodelgeneration,whichisusedtopredicttheeffort(orerrors)oftheprojectsinthetestdata.ThepredictioncriteriapresentedinSubsection5.1arethenusedtoconfirmwhethertheeffortwere 239September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookPredictingEffortandErrorsbyUsinganANN225accuratelypredictedornotby.Bothdatasets,thatis,thetrainingdataandtestdata,aredividedinto10sectionsandtheseareusedtorepeattheexperiment10times.5.3ResultsandDiscussionAtotalof1828projectsareusedintheexperiment,then914projectsareusedastestdataforeachexperiment.Foreachmethod,averagesoftheexperimentsresultsforthe10experimentsareshowninTable4.Table4ExperimentalResultsforEffortsPredictionMAESDAEMRESDREANNModel22.22399.4030.177820.96512MRAModel107.61213.002.04014.2476Table5ExperimentalResultsforErrorsPredictionMAESDAEMRESDREANNModel21.73893.0160.209261.1854MRAModel110.62237.372.63738.16175.3.1ValidationanalysisoftheaccuracyofthemodelsWecomparetheaccuracyoftheANNmodelwiththatoftheregressionanalysismodelusingWelch’st-test[Welch(1947)].Thet-test(calledStu-dent’st-test)[Student(1908)]isusedasatestofthenullhypothesisthatthemeansoftwonormallydistributedpopulationsareequal.Welch’st-testisusedwhenthevariancesoftwosamplesareassumedtobedifferenttotestthenullhypothesisthatthemeansofnontwonormallydistributedpopulationsareequalifthetwosamplesizesareequal[Aoki(2007)].Thetstatistictotestwhetherthemeansaredifferentiscalculatedasfollows:X−Yt0=√(14)sx+synxnywhereXandYarethesamplemeans,sxandsyarethesamplestandarddeviationsandnxandnyarethesamplesizes.Foruseinsignificance 240September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book226ReliabilityModelingwithApplicationstesting,thedistributionoftheteststatisticisapproximatedasanordinaryStudent’st-distributionwiththefollowingdegreesoffreedom:()2sx+synxnyν=(15)s2s2x+yn2(nx−1)n2(ny−1)xyThusoncetheat-valueanddegreesoffreedomhavebeendetermined,ap-valuecanbefoundusingatableofvaluesfromtheStudent’st-distribution.Ifthep-valueissmallerthanorequaltothesignificancelevel,thenthenullhypothesisisrejected.Thesignificancelevelsareusually0.05and0.01,arerepresentedbytheGreeksymbol,α.Thenullhypothesis,inthesecases,is“thereisnodifferencebetweenthemeansofthepredictionerrorsfortheANNmodelandtheMRAmodel”.Theresultsofthet-testforabsoluteerrorsandrelativeerrorstopredicttheamountofeffortaregiveninTables6and7,respectively.Andtheresultsofthet-testforabsoluteerrorsandrelativeerrorstopredictthenumberoferrorsaregiveninTables8and9,respectively.Theresultsindicatethatthemeansoftheabsolute(orrelative)er-rorsbetweenANNmodelsandMRAmodelshowsastatisticallysignificantdifference,becausethep-valuesarelessthan0.01.Table6Resultsoft-testforMAEforEffortANNModelMRAModelMean(X)22.223107.61Standarddeviation(s)99.403213.00Samplesize(n)91409140Degreesoffreedom(ν)12939.51tvalue(t0)34.7295pvalue<2.2×10 16Table7Resultsoft-testforMREforEffortANNModelMRAModelMean(X)0.177822.0401Standarddeviation(s)0.965124.2476Samplesize(n)91409140Degreesoffreedom(ν)10080.13tvalue(t0)40.9737pvalue<2.2×10 16 241September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookPredictingEffortandErrorsbyUsinganANN227Table8Resultsoft-testforMAEforErrorsANNModelMRAModelMean(X)21.738110.62Standarddeviation(s)93.016237.37Samplesize(n)91409140Degreesoffreedom(ν)11881.02tvalue(t0)33.3305pvalue<2.2×10 16Table9Resultsoft-testforMREforErrorsANNModelMRAModelMean(X)0.209262.6373Standarddeviation(s)1.18548.1617Samplesize(n)91409140Degreesoffreedom(ν)9524.393tvalue(t0)7.7881pvalue<2.2×10 166ConclusionInthischapter,wehaveestablishedeffortanderrorspredictionmodelsusingartificialneuralnetworks.Inaddition,wecarriedoutanevaluationexperimentthatcomparedtheaccuracyoftheANNmodelwiththatoftheMRAmodelusingWelch’st-test.TheresultsofthecomparisonindicatethattheANNmodelismoreaccuratethantheMRAmodel,becausethemeanerrorsoftheANNarestatisticallysignificantlylower.Ourfutureworksarethefollowing:1)Inthisstudy,weusedabasicartificialneuralnetwork.Morecomplexmodelsneedtobeconsideredtoimprovetheaccuracybyavoidingover-training.2)Weimplementedamodeltopredictthefinalamountofeffortandnum-beroferrorsinnewprojects.Itisalsoimportanttopredicteffortanderrorsmid-wayinthedevelopmentprocessofaproject.3)Weusedallthedatainimplementingthemodel.However,thedataincludeexceptionsandthereareharmfultothemodel.Dataneedstobeclusteredinordertotoidentifytheseexceptions. 242September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book228ReliabilityModelingwithApplicationsAcknowledgementsMydeepestappreciationgoestoProf.ToyoshiroNakashimawhosecom-mentsandsuggestionswereinnumerablyvaluablethroughoutthecourseofmystudy.IwouldalsoliketothankMr.YoshiyukiAnanandProf.NaohiroIshiiwhosecommentsmadeenormouscontributiontomywork.ReferencesAkaike,H.(1973).Informationtheoryandanextentionofthemaximumlike-lihoodprinciple,2ndInternationalSymposiumonInformationTheory,Petrov,B.N.,andCsaki,F.(eds.),pp.267–281.Aoki,S.(2007).Intestingwhetherthemeansoftwopopulationsaredifferent(inJapanese),http://aoki2.si.gunma-u.ac.jp/lecture/BF/index.html.Boehm,B.(1976).Softwareengineering,IEEETrans.SoftwareEng.C-25,12,pp.1226–1241.Hirayama,M.(2004).Currentstateofembeddedsoftware(injapanese),JournalofInformationProcessingSocietyofJapan(IPSJ)45,7,pp.677–681.Iwata,K.,Anan,Y.,Nakashima,T.andIshii,N.(2006).Improvingaccuracyofmultipleregressionanalysisforeffortpredictionmodel,Proceedingsof5thIEEE/ACISInternationalConferenceonComputerandInformationScience–ICIS2006,pp.48–55.Komiyama,T.(2003).Developmentoffoundationforeffectiveandefficientsoft-wareprocessimprovement(injapanese),JournalofInformationProcessingSocietyofJapan(IPSJ)44,4,pp.341–347.N.,U.(2004).Modelingtechniquesfordesigningembeddedsoftware(injapanese),JournalofInformationProcessingSocietyofJapan(IPSJ)45,7,pp.682–692.Nakamoto,Y.,Takada,H.andTamaru,K.(1997).Currentstateandtrendinembeddedsystems(injapanese),JournalofInformationProcessingSocietyofJapan(IPSJ)38,10,pp.871–878.Nakashima,S.(2004).Introductiontomodel-checkingofembeddedsoftware(injapanese),JournalofInformationProcessingSocietyofJapan(IPSJ)45,7,pp.690–693.Nakashima,T.,Iwata,K.,Anan,Y.andIshii,N.(2006).Studiesonprojectman-agementmodelsforembeddedsoftwaredevelopmentprojects,Proceedingsof4thInternationalConferenceonSoftwareEngineeringResearch,Man-agementApplications–SERA2006,pp.363–370.Ogasawara,H.andKojima,S.(2003).Processimprovementactivitiesthatputimportanceonstaypower(injapanese),JournalofInformationProcessingSocietyofJapan(IPSJ)44,4,pp.334–340.Student(1908).Theprobableerrorofamean,Biometrika6,1,pp.1–25.Takagi,Y.(2003).Acasestudyofthesuccessfactorinlarge-scalesoftwaresys-temdevelopmentproject(injapanese),JournalofInformationProcessingSocietyofJapan(IPSJ)44,4,pp.348–356. 243September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookPredictingEffortandErrorsbyUsinganANN229Tamaru,K.(2004).Trendsinsoftwaredevelopmentplatformforembed-dedsystems(injapanese),JournalofInformationProcessingSocietyofJapan(IPSJ)45,7,pp.699–703.Watanabe,H.(2004).Productlinetechnologyforsoftwaredevelopment(injapanese),JournalofInformationProcessingSocietyofJapan(IPSJ)45,7,pp.694–698.Welch,B.L.(1947).Thegeneralizationofstudent’sproblemwhenseveraldiffer-entpopulationvariancesareinvolved,Biometrika34,28. 244May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank 245September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookChapter13OptimalCheckpointTimesforDatabaseSystemsKenichiroNaruse1andSayoriMaeji21NagoyaSangyoUniversity,3255-5Arai-cho,Owariasahi488-8711,Japan2InstituteofConsumerScienceandHumanLife,KinjoGakuinUniversity,1723Oomori2,Moriyama,Nagoya463-8521,Japan1IntroductionMostcomputersystemsinofficesandindustriesexecutesuccessivelytaskseachofwhichhasrandomprocessingtimes.Insuchsystems,someerrorsoftenoccurduetonoises,humanerrorsandhardwarefaults.Todetectandmaskerrors,someusefulfaulttolerantcomputingtechniqueshavebeenadopted[LeeandAnderson(1990);SiewiorekandSwarz(eds)].Severalstudiesofdecidingoptimalcheckpointfrequencieshavebeenmade:Theperformanceandreliabilityofredundantmodularsystemswereevaluated[PradhanandVaidya(1992);Nakagawa(2008)],andtheperformanceofcheckpointschemeswithtaskduplicationwasevaluated[ZivandBruck(1997,1998)].Theoptimalinstruction-retryperiodthatminimizestheprobabilityofthedynamicfailurebyatriplemodularcontrollerwasderived[KimandShin(1996)].Theevaluationmodelswithfinitecheckpointsandboundedrollbackwerediscussed[Oharaetal.(2006-09-01)].Furthermore,checkpointingschemeforasetofmultipletasksinreal-timesystemswereinvestigated[ZhangandChakrabarty(2004)].Inrecentyears,internetshoppingisgreatlyincreasingintheworld.Manycompanieshavetoprocessbigdatawhichhavebeenusedforiden-tifyusers,shoppingitem,shoppinghistory,andsoon.Thesecomputers231 246September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book232ReliabilityModelingwithApplicationsneedtoprocessbigdatawithhi-speedresponse,andsometimes,maybesloweddownbyprocessingbigdata.Topreventsuchprocessing,wehavetostorethedataonmemory.Becausethememorycanreadandwriteover100timesfasterthanHDD,thedatabase-nodescanprocessbigdatawithhi-speed.Thischapterappliesthecheckpointoperationtothesystemwhichexe-cutessuccessivelyeachtaskwithrandomworkingtimes[Chenetal.(2010);Nakagawaetal.(2009);Naruseetal.(2006)]andaclusteringdatabasesystem.Section2considersthreeschemesofcheckpointinwhichtwoinde-pendentmodulesisexecutedandcomparetwostatesatcheckpointtimes.Iftwostatesofeachmoduledonotmatchwitheachother,wegobacktothenewestcheckpointandmaketheirretrialsinthreetypesofcheckpointschemes.Whenfailuresoccurinthesystem,itexecutestherecoveryoper-ationuntilthelatestcheckpointandrepeatssuchprocessesuntilthenextcheckpoint.Then,introducingcheckingcostsandalosscostfromafail-uretothelatestcheckpoint,thetotalexpectedcostbetweencheckpointsisobtained,usinganinspectionpolicy.Section3considersaclusteringdatabase-nodesystem[ORACLE(2012)]usinganextendedScheme3.Thesystemworksforseveralcomputerstoprocessbigdatawithhi-speedre-sponse.Thebigdataaredividedandstoredinndatabase-nodesonmem-orywhichcanreadandwriteover100timesfasterthanHDD.Inthiscase,thedatabase-nodescanprocessbigdatawithhi-speed.Wederiveopti-malcheckpointtimesandnumberofdatabase-nodes,andcomputebothoftheminnumericalexamples.Finally,weconsidertwotypesofdatabase-nodemodelssuchassinglenodeandmajoritynode[Nakagawa(2008)].Finally,Sect.4summarizestheresultsinthischapter.2RandomCheckpointModelsWeconsiderthreecheckpointschemeswhichhavedifferentcheckpointtypesandpresentnumericalexamplesforthreeschemes.SupposethatwehavetoexecutethesuccessivetaskswithaprocessingtimeYk(k=1,2,···)(Fig.1).Adoubledatabase-nodessystemoferrordetectionfortheprocessingofeachtaskisadopted.Then,introducingtwotypesofcheckpoints;compare-and-storecheckpoint(CSCP)andcompare-checkpoint(CCP)[Nakagawaetal.(2003)],weconsiderthefollowingthreecheckpointschemes:1)CSCPisplacedateachendoftasks(Fig.1), 247September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookOptimalCheckpointTimesforDatabaseSystems2332)CSCPisplacedattheN-th(N=1,2,···)endoftasks(Fig.2),3)CCPisplacedateachendoftasksandCSCPisplacedattheN-thendoftasks(Fig.3).Fig.1TaskexecutionforScheme1Fig.2TaskexecutionforScheme2Fig.3TaskexecutionforScheme3Themeanexecutiontimesperonetaskforeachschemeareobtained,andoptimalnumbersN∗thatminimizethemforSchemes2and3arederivedanalyticallyandarecomparednumerically.Thisisoneofappliedmodelswithrandommaintenancetimes[Nakagawa(2005);Sugiuraetal.(2004)]tocheckpointmodels.Suchschemeswouldbeusefulwhenitisbettertoplacecheckpointsattheendoftasksthanthoseonone’sway. 248September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book234ReliabilityModelingwithApplications2.1PerformanceAnalysisSupposethattaskkhasaprocessingtimeYk(k=1,2,···)withanidentical∫∞distributionG(t)≡Pr{Yk≤t}andafinitemeanµ=[1−G(t)]dt<0∞,andisexecutedsuccessively.Todetecterrors,weprovidetwoindepen-dentdatabase-nodeswheretheycomparetwostatesatcheckpointtimes.Further,itisassumedthatsomeerrorsoccurataconstantrateλ(λ>0),i.e.,theprobabilitythattwodatabase-nodeshavenoerrorduring(0,t]ise−2λt.(1)Scheme1CSCPisplacedateachendoftaskk:Whentwostatesofdatabase-nodesmatchwitheachotherattheendoftaskk,theprocessoftaskkiscorrectanditsstateisstored(Fig.1).Inthiscase,twodatabase-nodesgoforwardandexecutetaskk+1.However,whentwostatesdonotmatch,itisjudgedthatsomeerrorshaveoccurred.Then,twodatabase-nodesgobackandmaketheretryoftaskkagain.LetCbetheoverheadforthecomparisonoftwostatesandCsbetheoverheadfortheirstore.Then,themeanexecutiontimeoftheprocessoftaskkisgivenbyarenewalequation:∫∞{()[]}Le(1)=e−2λt(C+Cs+t)+1−e−2λtC+t+Le(1)dG(t).110(1)Solving(1)forLe1(1),C+µ+CsG∗(2λ)Le1(1)=,G∗(2λ)whereG∗(s)istheLaplace-Stieltjes(LS)transformofG(t),i.e.,G∗(s)≡∫∞e−stdG(t)fors>0.Therefore,themeanexecutiontimeperonetask0isC+µL1(1)≡Le1(1)=+Cs.(2)G∗(2λ)(2)Scheme2CSCPisplacedonlyattheendoftaskN(Fig.2):Whentwostatesofalltaskk(k=1,2,···,N)matchattheendoftaskN,itsstateisstoredandtwodatabase-nodesexecutetaskN+1.Whentwostatesdonotmatch,twodatabase-nodesgobacktothefirsttask1andmaketheirretries.Bythemethodsimilartoobtaining(1),themeanexecutiontimeoftheprocess 249September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookOptimalCheckpointTimesforDatabaseSystems235ofalltaskk(k=1,2,···,N)is∫∞{Le(N)=e−2λt(NC+Cs+t)20()[]}+1−e−2λtNC+t+Le(N)dG(N)(t),(3)2whereG(N)(t)istheN-foldStieltjesconvolutionofG(t)withitself,i.e.,∫tG(N)(t)≡G(N−1)(t−u)dG(u)(N=1,2,···),andG(0)(t)≡1for0t≥0andG(1)(t)=G(t).Solving(3)forLe(N),2NC+Nµ+Cs[G∗(2λ)]NLe2(N)=.[G∗(2λ)]NTherefore,themeanexecutiontimeperonetaskisLe2(N)C+µCsL2(N)≡=+(N=1,2,···).(4)N[G∗(2λ)]NNWhenN=1,L2(1)agreeswithL1(1)in(2).WefindanoptimalnumberN∗thatminimizesL(N).Thereexistsa22finiteN∗(1≤N∗<∞)becauselimL(N)=∞.Fromtheinequality22N→∞2L2(N+1)−L2(N)≥0,N(N+1)[1−G∗(2λ)]Cs≥(N=1,2,···).(5)[G∗(2λ)]N+1C+µTheleft-handsideof(5)isstrictlyincreasingto∞inN.Thus,thereexistsafiniteanduniqueminimumN∗(1≤N∗<∞)whichsatisfies(5).22If[1−G∗(2λ)]Cs≥,[G∗(2λ)]22(C+µ)thenN∗=1.2WhenG(t)=1−e−t/µ,(5)isrewrittenasNCsN(N+1)2λµ(2λµ+1)≥.(6)C+µExample13.1.Table1presentstheoptimalnumberN∗andtheresultingexecutiontime2L(N∗)/µandL(1)/µin(2)forλµandC/µwhenCs/µ=0.1.This221indicatesthatoptimalN∗decreasewithλµandincreasewithC/µ.For2example,whenλµ=0.005andC/µ=0.1,N∗=3andL(N∗)/µis1.167222thatisabout4%shorterthanL1(1)/µ=1.211forScheme1. 250September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book236ReliabilityModelingwithApplicationsTable1OptimalnumberNandtheresultingexecutiontimeL2(N)/µforScheme222andL1(1)/µforScheme1whenCs/µ=0.1.C/µ=0.5C/µ=0.1()()λµNL2N/µL1(1)/µNL2N/µL1(1)/µ22220.111.9001.90011.4201.4200.0511.7501.75011.3101.3100.0121.6111.63021.1941.2220.00531.5791.61531.1671.2110.00161.5351.60371.1301.2020.000581.5251.602101.1211.2010.0001181.5111.600211.1091.200(3)Scheme3CSCPisplacedattheendoftaskNandCCPisplacedonlyattheendoftaskk(k=1,2,···,N−1)betweenCSCPs(Fig.3):Whentwostatesoftaskk(k=1,2,···,N−1)matchattheendoftaskk,twodatabase-nodesexecutetaskk+1.Whentwostatesoftaskk(k=1,2,···,N)donotmatch,twodatabase-nodesgobacktothefirsttask1.WhentwostatesoftaskNmatch,theprocessofalltasksNiscompleted,anditsstateisstored.Twodatabase-nodesexecutetaskN+1.LetLe4(k)bethemeanexecutiontimefromtaskktothecompletionoftaskN.Then,bythemethodsimilartoobtaining(3),∫∞{[]()[]}Le(k)=e−2λtC+t+Le(k+1)+1−e−2λtC+t+Le(k+1)dG(t)4440(k=1,2,···,N−1),∫∞{()[]}Le(N)=e−2λt(C+t+Cs)+1−e−2λtC+t+Le(1)dG(t).440(7)Solving(7)forLe4(1),{}(C+µ)1−[G∗(2λ)]NLe4(1)=+Cs.[1−G∗(2λ)][G∗(2λ)]NTherefore,themeanexecutiontimeperonetaskis{}(C+µ)1−[G∗(2λ)]NLe4(1)CsL4(N)≡=+(N=1,2,···).NN[1−G∗(2λ)][G∗(2λ)]NN(8) 251September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookOptimalCheckpointTimesforDatabaseSystems237WhenN=1,L4(1)agreeswith(2).Bycomparing(4)with(8),Scheme3isbetterthanScheme2.ItcanbeclearlyseenthatafiniteN∗(1≤N∗<∞)thatminimizes44L4(N)exists.FromtheinequalityL4(N+1)−L4(N)≥0,1∑N{}Cs∗j1−[G(2λ)]≥(N=1,2,···),(9)[G∗(2λ)]N+1C+µj=1whoseleft-handsideisstrictlyincreasingto∞inN.Thus,thereexistsafiniteanduniqueminimumN∗(1≤N∗<∞)whichsatisfies(9).If44(C+µ)[1−G∗(2λ)]≥Cs[G∗(2λ)]2,thenN∗=1.Bycomparing(9)with4(5),itcanbeeasilyseenthatN∗≥N∗.43WhenG(t)=1−e−t/µ,(9)isrewrittenasN[()j]N+1∑1Cs/µ(2λµ+1)1−≥.(10)2λµ+1C/µ+1j=1Example13.2.Table2presentstheoptimalnumberN∗andtheresultingexecutiontime4L(N∗)/µin(8)forλµandC/µwhenCs/µ=0.1.Clearly,Scheme3is44betterthanScheme2andN∗≥N∗.However,ingeneral,theoverheadC43forScheme2wouldbelessthanthatforScheme3.Insuchcase,Scheme2mightbebetterthanScheme3.Table2OptimalnumberNandtheresultingexecutiontimeL4(N)/µforScheme344whenCs/µ=0.1.C/µ=0.5C/µ=0.1λµ()()NL4N/µNL4N/µ44440.111.90011.4200.0511.75011.3100.0131.59431.1780.00541.56341.1530.00181.52691.1220.0005121.518131.1150.0001261.508301.107 252September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book238ReliabilityModelingwithApplications3Database-nodeModelsWeconsidertwotypesofdatabase-nodemodelssuchasasingledatabase-nodemodel(Fig.4)andamajoritydatabase-nodemodel(Fig.5),andpresentnumericalexamplesforbothofthem.Fig.4Singlenodemodel.Fig.5Majoritynodemodel.(1)SinglenodeWeconsideronemanager-nodeandndatabase-nodesinadatabasesystem.Thedatabase-nodesprocesseachtask,andthemanager-nodewatchesthedatabase-nodesinFig.4.Firstofall,thedatabase-nodesstoredatabasedatafromHDDtomemorybydividingthemintothenumberofdatabase-nodes,andstoreapartofdataonmemory.Sothat,thedatabase-nodescanprocesseachtaskwithhi-speed.However,ifsomeerrorsoccurinthedatabase-nodes,thedataislostonmemory.Topreventsuchdataloss,wesupposethefollowingcheckpointschemessuchasJournalCheckpoint(JC)andFlushCheckpoint(FC)schemes[ORACLE(2012)]:JCisplacedateachendoftasksandFCisplacedattheNthendoftasks.JCstorescommandandresultofatasktoHDD,andFCupdatestheHDDdatabase 253September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookOptimalCheckpointTimesforDatabaseSystems239accordingtoJCdata.Themanagement-nodewatchesndatabase-nodes.Ifsomeerrorsoccurinthedatabase-nodes,themanagement-noderestartsthemandtransportsapartofcorrespondingdatafromHDDtomemory.ThischeckpointschemeisthesameasScheme3ofSect.2exceptthatScheme3comparetwodatabase-noderesultstodetectfailures,butthisschemecandetectfailuresbymanager-node.(2)MajoritynodeWeextendasinglenodemodeltoaredundantdatabase-nodemodelwhichconsistsofngroupsof(2m+1)database-nodesandonemanager-nodeinFig.5.Theredundantmodelisamajoritydecisionsystemwith2m+1database-nodesasanerrormaskingsystem,i.e.,(m+1)-out-of-(2m+1)system[Nakagawa(2008)].Ifmorethanm+1resultsof2m+1database-nodesagree,themanager-nodejudgesthedatabase-nodeiscorrect,andthedatabase-nodecanexecutethenexttask.Thus,iflessthanmerrorsoccurinthegroupof2m+1database-nodes,thedatabase-nodescanmasktheerrorusingoneofamajoritydecisionpolicy.Ifmorethanm+1er-rorsoccurintheonedatabase-nodes,themanagement-noderestartsthedatabase-nodeandtransportsapartofcorrespondingdatafromHDDtomemory.3.1PerformanceAnalysisLetCbetheoverheadforJCandCsbetheoverheadforFCwithC 254September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book240ReliabilityModelingwithApplicationsnodeexecutestaskN+1.ItisassumedF(t)isafailureprobabilityofdatabase-node.LetLe(n,k)bethemeanexecutiontimefromtaskktothecompletionoftaskN.Then,bythesimilarmethodofobtaining(2)[Maejietal.(2010)],∫∞{[]Le(n,k)=F(t)C+t+Le(n,k+1)0[]}W+F(t)C+t+Le(n,1)++RdG(t)(k=1,2,···,N−1),n∫∞{Le(n,N)=F(t)[C+t+Cs]0[]}W+F(t)C+t+Le(n,1)++RdG(t),(11)nwhereF(t)≡1−F(t).(1)SinglenodeItisassumedthatsomeerrorsoccurataconstantrateλ(λ>0),i.e.,theprobabilitythatndatabase-nodeshavenoerrorduring(0,t]isF(t)=e−nλt(n=1,2,···).Solving(11)forLes(n,1),(){}C+µ+W+R1−[G∗(nλ)]NnLes(n,1)=+Cs.(12)[1−G∗(nλ)][G∗(nλ)]NTherefore,themeanexecutiontimeperonetaskis(){}C+µ+W+R1−[G∗(nλ)]NLes(n,1)nCsLs(n,N)≡=+NN[1−G∗(nλ)][G∗(nλ)]NN(n,N=1,2,···).(13)WhenG(t)=1−e−t/µ,(13)isrewrittenas()C+µ+W+R∑NnjCsLs(n,N)=(nλµ+1)+.(14)NNj=1WefindanoptimalnumberN∗thatminimizesL(n,N)in(14)forsn≥1.FromtheinequalityLs(n,N+1)−Ls(n,N)≥0,N[()j]Cs∑1(nλµ+1)N+11−≥µnλµ+1C+1+W/n+Rj=1µµ(N=1,2,···).(15) 255September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookOptimalCheckpointTimesforDatabaseSystems241Clearly,ifCsµnλµ(nλµ+1)≥,C+1+W/n+RµµthenN∗=1.Clearly,N∗decreaseswithn.Example13.3.Table3presentsanoptimalnumberN∗in(15)andtheresultingexecutiontimeL(n,N∗)/µin(14)forλµandnwhenCs/µ=0.8,C/µ=0.2,sW/µ=5andR/µ=10.ThisindicatesthatN∗decreasewithbothnandλµ,anditsresultingmeantimesL(n,N∗)decreasewithnandincreaseswithλµforN∗≥2.ThecaseofN∗=1meansthatweshouldprovidenoJCbetweenFC.Forexample,whenλµ=0.005andn=2,wemake3JCcheckpointsbetweenFCcheckpoints.Inthiscase,theexecutiontimeis14.242.Table3OptimalnumberNandresultingexecutiontimeLs(n,N)/µforλµandnwhenCs/µ=0.8,C/µ=0.2,W/µ=5andR/µ=10.n=1n=2n=5n=10λµL∗∗∗∗Ns(n;N)NLs(n;N)NLs(n;N)NLs(n;N)0.1118.620117.240119.100124.2000.05117.810115.870116.050118.3500.01316.793214.514213.530113.6700.005416.604314.242213.061212.9920.0011016.369813.924512.544412.1950.00051416.3181113.855712.437512.0370.00013116.2522413.7681612.3021211.8430.000054416.2363413.7482312.2711711.800Next,wefindanoptimaln∗thatminimizesL(n,N)in(14)forN≥1.sFromtheinequalityLs(n+1,N)−Ls(n,N)≥0,()NW∑jC+µ++R[(n+1)λµ+1]n+1j=1()NW∑j≥C+µ++R(nλµ+1)(n=1,2,···).(16)nj=1Especially,whenN=1,(16)isWn(n+1)≥,(17)C+µ+R 256September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book242ReliabilityModelingwithApplicationsandwhenN=2,2W(C+µ+R)λµ[(2n+1)λµ+3]+W(λµ)≥.(18)n(n+1)Theleft-handsideof(18)isstrictlyincreasingto∞,andtheright-handsideisdecreasingfromW/2to0.Thus,anoptimalnumbern∗(1≤n∗<∞)whichsatisfy(18)existsuniquely.If2W3λµ(λµ+1)(C+µ+R)+W(λµ)≥,2thenn∗=1.Example13.4.Table4presentsanoptimalnumbern∗in(16)andtheresultingexecutiontimeL(n∗,N)/µin(14)forλµandNwhenCs/µ=0.8,C/µ=0.2,sW/µ=5andR/µ=10.TheresultingexecutiontimesL(n∗,N)increaseswithNandλµ.ThisshowsasimilartendencytoTable1.Table4OptimalnumbernandresultingexecutiontimeLs(n,N)/µforλµandNwhenCs/µ=0.8,C/µ=0.2,W/µ=5andR/µ=10.N=1N=2N=5N=10λµL∗∗∗∗ns(n;N)nLs(n;N)nLs(n;N)nLs(n;N)0.1217.240218.484121.919128.4800.05315.597216.224218.561121.4750.01713.548513.530414.186315.2730.005913.085812.944513.306413.9850.0012112.4781712.1891212.202912.4330.00053012.3372412.0151711.9511312.0870.00016712.1505411.7843811.6212811.6350.000059412.1067711.7305411.5444011.530Finally,wefindbothoptimaln∗andN∗thatminimizeL(n,N)in(13).sTocomputebothoptimaln∗andN∗,wesubstituten=1into(15),andcomputen∗forN=N∗in(16),andcomputeN∗forn=n∗into(15).If1121N∗=N∗andn∗=n∗thenN∗=N∗andn∗=n∗.Bothn∗andN∗II+1II+1IIdecreasewithλµ.Thisindicatesanoptimalcombinationofthenumbersofdatabase-nodesandJCbetweenFC.Table5presentsoptimalnumbern∗,N∗andtheresultingexecutiontimeL(n∗,N∗)/µwhenCs/µ=0.8,C/µ=0.2,W/µ=5andR/µ=10.sThisindicatesn∗/N∗decreasegraduallywithλµ,andL(n∗,N∗)increases 257September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookOptimalCheckpointTimesforDatabaseSystems243Table5Optimalnumbern,NandtheresultingexecutiontimeLs(n,N)/µforλµwhenCs/µ=0.8,C/µ=0.2,W/µ=5andR/µ=10.L∗∗λµnNs(n;N)0.12117.2400.053115.5970.015213.5300.0058212.9440.00115312.1490.000519411.9380.000133711.6170.0000544811.526withλµ.Itcanbeeasilyseenthatthepairof(n∗,N∗)isbetterthanany(n,N∗)and(n∗,N)forthesameλµinTables3and4.(2)MajoritynodemodelWeconsidera2-out-of-3database-nodesystem.Then,theprobabilitythatonedatabase-nodeiscorrectduring(0,T]is[Nakagawa(2008);Naruseetal.(2006)]F(t)=3e−2λt−2e−3λt.(19)Thus,theprobabilitythatndatabase-nodesystemsarecorrectis[]n(−2λt−3λt)nF(t)=3e−2e.(20)From(13),themeanexecutiontimeperonetaskis(){[∫∞()n]N}C+µ+W+R1−3e−2λt−2e−3λtdG(t)n0CsLm(n,N)=[∫∞−2λt−3λtn][∫∞−2λt−3λtn]N+NN1−(3e−2e)dG(t)(3e−2e)dG(t)00(n,N=1,2,···).(21)WhenG(t)=1−e−t/µ,(21)is()∑C+µ+W+RNA−jnj=1CsLm(n,N)=+,(22)NNwhere()nin−iin(−1)32∑iA≡.(2n+i)λµ+1i=0 258September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book244ReliabilityModelingwithApplicationsWefindanoptimalnumberN∗for2-out-of-3nodemodelthatminimizesLm(n,N)in(22)forn≥1.Example13.5.Table6presentsoptimalnumberN∗andtheresultingexecutiontimeL(n,N∗)/µforλµandnwhenCs/µ=0.8,C/µ=0.2,W/µ=5,mR/µ=10andµ=1.ThisshowsasimilartendencytoTables3and4,andthemeanexecutiontimeisalittleshorterthanasinglenodemodel.Table6OptimalnumberNandresultingexecutiontimeLm(n,N)/µforλµandnwhenCs/µ=0.8,C/µ=0.2,W/µ=5,R/µ=10andµ=1.n=1n=2n=5n=10λµL∗L∗L∗L∗Nm(n;N)Nm(n;N)Nm(n;N)Nm(n;N)0.1217.585115.554115.131116.1280.05316.859214.589213.654113.7450.011316.3261013.866712.454512.0600.0052616.2632013.7821312.3241011.8740.00112916.2129913.7166612.2244811.7340.000525716.20619813.70813212.2129611.7170.0001128316.20198713.70266112.20247711.7030.00005256616.201197313.701132212.20198111.702Next,wefindanoptimalnumbern∗for2-out-of-3nodemodelthatminimizesLm(n,N)in(22)forN≥1.Example13.6.Table7presentsanoptimalnumbern∗andtheresultingexecutiontimeL(n∗,N)/µforλµandnwhenCs/µ=0.8,C/µ=0.2,W/µ=5,R/µ=m10andµ=1.TheresultingexecutiontimesL(n∗,N)increasewithNmandλµ.ThisshowsasimilartendencytoTables1,2and4.Finally,wefindbothoptimaln∗andN∗thatminimizeL(n,N)inm(22).Wecomputebothoptimaln∗andN∗bythesimilarmethodusedinScheme3ofSect.2.Thisindicatesanoptimalcombinationofnumbersofdatabase-nodesandJCbetweenFCfora2-out-of-3nodemodel.Example13.7.Table8presentsoptimalnumbern∗,N∗andtheresultingexecutiontimeL(n∗,N∗)/µwhenCs/µ=0.8,C/µ=0.2,W/µ=5,R/µ=10andm 259September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookOptimalCheckpointTimesforDatabaseSystems245Table7OptimalnumbernandresultingexecutiontimeLm(n,N)/µforλµandnwhenCs/µ=0.8,C/µ=0.2,W/µ=5,R/µ=10andµ=1.N=1N=2N=5N=10λµL∗;N)L∗;N)L∗;N)L∗;N)nm(nnm(nnm(nnm(n0.1415.041315.496217.365120.3080.05713.633513.654414.432215.6920.012912.3562312.0391611.9871212.1390.0055612.1804611.8223211.6752311.7100.00127412.03722411.64515811.42311711.3660.000554712.01844611.62231511.39223311.3230.0001272912.004222811.604157511.366116311.2890.00005545712.002445511.602315011.363232611.284Table8Optimalnumbern,NandresultingexecutiontimeLm(n,N)/µforλµwhenCs/µ=0.8,C/µ=0.2,W/µ=5,R/µ=10andµ=1.L∗;N∗λµnNm(n)0.14115.0410.055213.6540.0118411.9710.00530611.6750.001941611.3570.00051512511.2980.00014487311.2330.0000571311611.221µ=1.Thisindicatesn∗/N∗decreasegraduallywithλµ,andL(n∗,N∗)mincreasewithλµandarelessthanthoseinTables6and7.4ConclusionThischapterhaveconsideredthreetypesofcheckpointschemesandtwotypesofdatabase-nodessystems,andhavesolvedtheproblemsinwhatplacesweshouldplacesuitablecheckpoints,provideoptimalnumberofdatabase-nodesandcomparewhichdatabase-nodemodelshowsgoodper-formance.WehavederivedtheoptimalJCnumberbetweenFCforthenumberofdatabase-nodes,theoptimaldatabase-nodesforthenumberoftasksforFC,theoptimaldatabase-nodesandJCnumberbetweenFC,andtheoptimalexecutiontimeforthesamenumberofdatabase-nodes.Ithasbeenshownthattheoptimalmeanexecutiontimesdecreasewhentherates 260September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book246ReliabilityModelingwithApplicationsµ/(1/λ)ofthemeanprocessingtimewiththemeanerrortimedecrease,andthatthemeantimeofexecutiontimeina2-out-of-3nodemodelissmallerthanasinglenodemodelinsomerange.Thiswouldbeappliedtocloudsystemsandotherdatabasesystemsrequestedforhighreliabilityandhighspeedprocessing.ReferencesChen,M.,Nakamura,S.andNakagawa,T.(2010).Replacementandpreven-tivemaintenancemodelswithrandomworkingtimes,IEICETRANS-ACTIONSonFundamentalsofElectronics,CommunicationsandCom-puterSciences93-A,2,pp.500–507,doi:10.1587/transfun.E93.A.500,http://ci.nii.ac.jp/naid/10026863281/.Kim,H.andShin,K.G.(1996).Designandanalysisofanoptimalinstruction-retrypolicyfortmrcontrollercomputers,IEEETransactionsonComputers45,11,pp.1217–1225.Lee,P.A.andAnderson,T.(1990).FaultTolerancePrinciplesandPractice(Springer,Wien).Maeji,S.,Naruse,K.andNakagawa,T.(2010).Optimalcheckingmodelswithrandomworkingtimes,inAdvancedReliabilityandModellingIV,pp.488–495.Nakagawa,S.,Fukumoto,S.andIshii,N.(2003).Optimalcheckpointingintervalsofthreeerrordetectionschemesbyadoublemodularredundancy,Mathe-maticalandComputingModeling38,11–13,pp.1357–1363.Nakagawa,T.(2005).MaintenanceTheoryofReliability(Springer,London).Nakagawa,T.(2008).AdvancedReliabilityModelsandMaintenancePolicies(Springer,London).Nakagawa,T.,Naruse,K.,andMaeji,S.(2009).RandomcheckpointmodelswithNtandemtasks,IEICETRANSACTIONSonFundamentalsofElectronics,CommunicationsandComputerSciencesE92-A,2,pp.1572–1577.Naruse,K.,Nakagawa,T.andMaeji,S.(2006).Optimalchecpointintervalsforerrordetectionbymultiplemodularredundancies,inAdvancedReliabilityandModellingII,pp.293–300.Ohara,M.,Suzuki,R.,Arai,M.,Fukumoto,S.andIwasaki,K.(2006-09-01).Analyticalmodelonhybridstatesavingwithalimitednum-berofcheckpointsandboundrollbacks(reliability,maintainabilityandsafetyanalysis),IEICETRANSACTIONSonFundamentalsofElec-tronics,CommunicationsandComputerSciences89,9,pp.2386–2395,http://ci.nii.ac.jp/naid/110007537954/.ORACLE(2012).Mysql::Mysqlclustercge,URLhttp://www-jp.mysql.com/products/cluster/.Pradhan,D.andVaidya,N.(1992).Roll-forwardcheckpointingscheme:Concur-rentretrywithnon-dedicatedspares,inProceedingsoftheIEEEWorkshoponFault-TolerantParallelandDistributedSystems,pp.166–174. 261September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookOptimalCheckpointTimesforDatabaseSystems247Siewiorek,D.P.andSwarz(eds),R.S.(1982).TheTheoryandPracticeofReliableSystemDesign(DigitalPress,Bedford,Massachusetts).Sugiura,T.,Mizutani,S.andNakagawa,T.(2004).Optimalrandomreplacementpolices,inTenthISSATInternationalConferenceonReliabilityandQualityinDesign,pp.99–103.Zhang,Y.andChakrabarty,K.(2004).Dynamicadaptationforfaulttoleranceandpowermanagementinembeddedreal-timesystems,ACMTransactionsonEmbeddedComputing3,2,pp.336–360.Ziv,A.andBruck,J.(1997).Performanceoptimizationofcheckpointingschemeswithtaskduplication,IEEETransactionsonComputers46,12,pp.1381–1386.Ziv,A.andBruck,J.(1998).Analysisofcheckpointingschemeswithtaskdupli-cation,IEEETransactionsonComputers47,2,pp.222–227. 262May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank 263September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookChapter14PeriodicandRandomInspectionsforaComputerSystemMingchihChen1,XufengZhao2andSyoujiNakamura31GraduateInstituteofBusinessAdministration,FuJenCatholicUniversity,No.510,ZhongzhengRd.,Xinzhuang,NewTaipeiCity24205,Taiwan2GraduateSchoolofManagementandInformationSciences,AichiInstituteofTechnology,1247Yachigusa,Yakusa-cho,Toyota470-0392,Japan3DepartmentofHumanLifeandInformation,KinjoGakuinUniversity,1723Omori2,Moriyama,Nagoya463-8521,Japan1IntroductionIthasbeenwell-knownthatfaultsincomputersystemssometimesoccurintermittently[MalaiyaandSu(1981);Castilloetal.(1982);Nakagawa(2005)]:Faultsarehiddenandbecomepermanentfailurewhenthedu-rationofhiddenfaultsexceedsathresholdlevel[Nakagawaetal.(1993);Nakagawa(2008)].Topreventsuchfaults,someinspectionpoliciesforcom-putersystemswereconsidered[MalaiyaandSu(1981);Suetal.(1978);Malaiya(1982)],datatransmissionstrategiesforcommunicationsystemswereconsidered[Yasuietal.(2002)],andsomepropertiesforsecuritymeasuresofsoftwareincomputersystemswereobserved[LiuandTraore(2007)].Thereliabilitymodelsandavarietyofmaintenancemodelsplayanim-portantroleinmanufacturingorcomputersystems.Someapplicationsofreliabilitymodelsincomputersystems,suchascommunications,backuppolicies,checkpointintervals,weresummarized[NakamuraandNaka-gawa(2010)].Thelatestworkproposednewreliabilityandfault-tolerant249 264September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book250ReliabilityModelingwithApplicationsmethodswhichwereappliedinmanufacturingmodules[SavsarandAldai-hani(2012)],systemreliabilityallocationbasedonBayesiannetwork[Qianetal.(2012)],supportingreal-timedataservices[Xiaoetal.(2012)],andsoftwarereliabilitymodelingbasedongeneexpression[ZhangandXiao(2012)].Somesystemsinofficesandindustriesexecutesuccessivejobsandcom-puterprocesses.Forsuchsystems,itwouldbeimpossibleorimpracticaltomaintaintheminastrictlyperiodicfashion[BarlowandProschan(1965)].Forexample,whenajobhasavariableworkingcycleandprocessingtime,itwouldbebettertodosomemaintenanceafterithascompleteditsworkandprocess[Nakagawa(2005);ZhaoandNakagawa(2012)].Theopti-malperiodicpolicyforsuchsystemswhentheworkingtimeisexponentialwasderived[Nakagawa(2005)]andperiodicandrandominspectionpoliciesweresummarized[Nakagawa(2008)].Furthermore,severalbackuppoliciesfordatabasesystemswithrandomworkingtimeswerediscussed[Naruseetal.(2009);Maejietal.(2010)],byapplyingtheinspectionpolicytothebackuppolicy.Wefirstlyapplyastandardinspectionpolicy[Nakagawa(2005)]withimperfectcheckstoacomputersysteminthischapter:Thesystemhastobeoperatedforaninfinitetimespan.Todetectfault,thesystemischeckedatperiodictimesandrandomtimesaccordingtocomputerprocesses,whicharecalledperiodicinspectionandrandominspection,respectively.Systemfaultisdetectedatthenextcheckingtimewithacertainprobabilityandundetectedfaultisdetectedatthesecondcheckingtimewiththesameprobability.Secondly,weapplyperiodicandrandominspectionsintofaultdetection,rollbackoperation,andbackupprocess[Reuter(1984);Fukumoto(1992)]foracomputersystem,whosewholeprocessesiscalledbackupoperationinthischapter.Twocaseswhenthefaultcouldbedetectedimmediatelyandatthefollowingcheckingtimearediscussed.Fortheabovetwomodels,expectedcostsareobtainedandtheiropti-malpolicieswhichminimizethemarederivedwhenfaultandprocesstimesareexponential.Comparisonsbetweenoptimaltimesforperiodicandran-dominspectionswhenbothfaultandprocessingtimesareexponentialaregiven.Itisshownthatperiodicinspectionisbetterthantherandompol-icy;however,ifamodifiedrandominspectioncostislowerthanthatforperiodicpolicy,thentwoinspectionsarealmostthesame.Furthermore,weconsiderrandominspectionpoliciesinwhichthesystemischeckedattheNthintervalofrandomprocesstimesforrespectivemodels. 265September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookInspectionsforaComputerSystem2512ModelI2.1PeriodicInspectionConsiderastandardinspectionpolicy[Nakagawa(2005)]withimperfectchecks:AcomputersystemshouldoperateforaninfinitetimespanandischeckedatperiodictimeskT(k=1,2,···).Systemfaultisdetectedatthefollowingcheckingtimewithprobabilityq(0 266September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book252ReliabilityModelingwithApplicationsTherefore,thetotalexpectedcostuntilmaintenanceis,from(1)and(2),∑∞1cDCP(T)=(cT+cDT)F(jT)+−.(3)qλj=1Inparticular,whenF(t)=1−e−λt(0<1/λ<∞),()1pcDCP(T)=(cT+cDT)1−e−λT+q−λ.(4)DifferentiatingCP(T)withrespecttoTandsettingitequaltozero,p−λT2λTλTλcT(1−e)e+e−1−λT=,(5)qcDwhoseleft-handsideincreasesfrom0to∞.Thus,thereexistsanoptimalT∗(0 267September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookInspectionsforaComputerSystem253∑∞∫∞[∫∞]=[F(t)−F(t+t)]dG(t)dG(j)(t)11221j=000∑∞∫∞∑∞∫∞=F(t)dG(j)(t)−F(t)dG(t)=1.j=00j=10Then,theexpectednumberofchecksuntilmaintenanceis∑∞∑∞∫∞[∫∞](k+1)pkq+j[F(t)−F(t)]dG(t−t)dG(j)(t)21211k=0j=00t1∫∞1=M(t)dF(t)+,(8)0q∑∞(j)whereM(t)≡G(t)denotestheexpectednumberofchecksin[0,t],j=1whichiscalledarenewalfunctioninstochasticprocesses(Nakagawa,2011).Themeantimefromfaulttoitsdetectionis∑∞∫∞∫∞∫t2dG(j)(t)dG(t−t)dF(t)121j=00t1t1∑∞∫∞×pkq(t−t)dG(k)(t−t)332k=0t2∑∞∫∞∫∞∫t2(p)=dG(j)(t)dG(t−t)+t−tdF(t)1212j=00t1t1qµp∑∞∫∞∫∞∫t1+t2=+dG(j)(t)dG(t)[F(t)−F(t)]dt121qµj=000t1∑∞∫∞∫∞=dG(j)(t)G(t)[F(t)−F(t+t)]dt111j=000∫∞p111+=+M(t)dF(t)−.(9)qµqµµ0λTherefore,thetotalexpectedcostuntilmaintenanceis,from(8)and(9),()[∫]∞cD1cDCR(G)=cR+M(t)dF(t)+−.µ0qλInparticular,whenG(t)=1−e−µt(0<1/µ<∞),i.e.,M(t)=µt,theexpectedcostisafunctionofµwhichisgivenby()()cDµ1cDCR(µ)=cR++−.(10)µλqλ 268September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book254ReliabilityModelingwithApplicationsDifferentiatingCR(µ)withrespecttoµandsettingitequaltozero,()2λqλcR=,(11)µcDandtheresultingcostis()C(µ∗)λλR=+2.(12)cD/λqµqµ2.3ComparisonofPeriodicandRandomInspectionsWecompareperiodicandrandominspectionpoliciestheoreticallyandnu-mericallywhenF(t)=1−e−λtandG(t)=1−e−µt.Forthesimplicityofnotations,itisassumedthatq=1,λ=1,cT=cR,andc≡λcT/cD≤1becauseexpectedlosscostforthemeanfaultdelayedtime1/λwouldbemuchhigherthancostcTofonecheckinmostinspectionmodels.Undertheaboveassumptions,thetotalexpectedcostfortheperiodicinspectionis,from(4),CP(T)c+T=−1.(13)cD1−e−TAnoptimalT∗whichminimizesC(T)isgivenbyasolutionofequationPeT−1−T=c,(14)andtheresultingcostis∗CP(T)∗T∗=c+T=e−1.(15)cDThetotalexpectedcostforrandominspectionis,from(10),CR(µ)1=c(1+µ)+.(16)cDµAnoptimalµ∗whichminimizesC(µ)isgivenbyR1√=c,(17)µ∗andtheresultingcostis∗()2CR(µ)∗112=c(1+µ)+=+.(18)cDµ∗µ∗µ∗Whenc=1,T∗=1.1462from(14),and1/µ∗=1from(17).Thus,itiseasilyprovedthat0 269September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookInspectionsforaComputerSystem255From(14)and(17),computeasolutionofequationQ(T)≡eT−(1+T+T2)=0.(19)Clearly,asolutionof(19)isabout1.79.Thus,Q(T)<0for0 270September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book256ReliabilityModelingwithApplicationsTable1OptimalT,1/µ,andCP(T)/cD,CR(µ)/cDwhenq=0.9andλ=1.cT/cDTCP(T)/cD1/µCR(µ)/cD0.0010.04030.05020.03000.06780.0020.05660.07140.04240.09650.0050.08930.11430.06710.15460.0100.12560.16390.09490.22190.0200.17640.23630.13420.32040.0500.27460.38790.21210.52700.1000.38240.57160.30000.77780.2000.52820.85560.42431.16500.5000.79941.50520.67082.04631.0001.07572.38070.94873.2193expectedcostsoftwoinspectionpoliciesarethesame.From(6)and(12),wecomputeµbwhichsatisfies[2]()2λT∗p−λT∗1λ2λ(e−1)(1−e)+p+1=+,qqµbµbandcompute()2bcR1λ=.cD/λqµbTable2presents1/µb,bcR/cDandbcR/cTforcT/cDwhenq=0.9andλ=1.ThisindicatesthatbcRisalittlemorethanthehalfofcT.Inotherwords,whencR≈cT/2,twoexpectedcostsforperiodicandrandominspectionsarealmostthesame.Table2Valuesof1/µb,bcR/cDandbcR/cTwhenq=0.9andλ=1.cT/cD1/µbbcR/cDbcR/cT0.0010.02240.00050.50390.0020.03180.00100.50540.0050.05040.00250.50860.0100.07150.00510.51180.0200.10160.01030.51600.0500.16210.02630.52530.1000.23130.05350.53520.2000.33120.10970.54850.5000.53550.28680.57351.0000.77380.59870.5987Itisnotedfrom(14)and(17)thatT∗→0and1/µ∗→0asc→0.Thus,from(15)and(18),eT−11lim=.T→0T2+2T2 271September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookInspectionsforaComputerSystem257Thisshowsthatifc→0,thenC(T∗)→C(µ∗)/2.Sothat,itwouldbeTRestimatedthatifc→0andcR/cT=0.5,thentwoexpectedcostsoftheperiodicandrandompolicieswouldbethesameasshowninTable2.2.4NthRandomInspectionSupposethatthecomputersystemischeckedattimesSjN(j=1,2,···,N=1,2,···),i.e.,attimesS1N,S2N,···.WhenN=1,thesystemischeckedateverySjinSection2.2.Then,fromCR(G),re-placingG(t)withG(N)(t),1/µwithN/µ,andM(t)withM(N)(t)=∑∞(jN)G(t)(N=1,2,···),thetotalexpectedcostuntilfaultdetec-j=1tionis()[∫]∞NcD(N)1cDCR(N)=cR+M(t)dF(t)+−µ0qλ(N=1,2,···).(20)Inparticular,whenF(t)=1−e−λt,∫∞∗N−λt(N)[G(λ)]edM(t)=,01−[G∗(λ)]N∫∞whereG∗(λ)≡e−λtdG(t),andtheexpectedcostin(20)is0()()NcD1pcDCR(N)=cR++−.(21)µ1−ANqλwhereA=G∗(λ)<1.FromtheinequalityC(N+1)−C(N)≥0,RR[]1−ANpc1+(1−AN+1)−N≥R,(22)(1−A)ANqcD/µwhichincreasesstrictlywithNto∞.Thus,thereexistsauniqueminimumN∗(1≤N∗<∞)whichsatisfies(22).Clearly,N∗increasesstrictlywithqfrom1toasolutionoftheequation1−ANcR−N≥.(1−A)ANcD/µNotethatwhenG(t)=1−e−µt,A=µ/(λ+µ).Table3presentsoptimalN∗andtheresultingcostC(N∗)/cfor1/µRDandc/cwhenq=0.9and1/λ=1.ThisindicatesthatoptimalN∗RDdecreaseswith1/µandincreaseswithc/c,andN∗/µarealmosttheRDsameforsmall1/µ.ComparedTable3withTable1,ifcT=cR,whenc/c=0.5,1/µ∗ 272September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book258ReliabilityModelingwithApplicationsTable3OptimalNandCR(N)/cDwhenq=0.9andλ=1.cR/cD=0.5cR/cD=1.01/µNCR(N)/cDNCR(N)/cD0.01801.51291082.38940.02401.5206542.39810.05161.5435222.42410.1081.5812112.46660.2041.655362.55220.5021.866722.82221.0012.166713.22223ModelII3.1PeriodicandRandomInspectionsWeapplytheaboveinspectionmethodsintorecoveryandbackuptech-niquesinacomputersystem.SupposethatacomputersystemischeckedatperiodictimeskT(k=1,2,···)foraspecifiedT>0andalsoatsucces-siverandomtimesSj(j=1,2,···)suchasworkingandprocessingtimes.Whenafaultoccurs,itisdetectedimmediately,androllbackoperationisexecuteduntilthelatestcheckingtime,andthenbackupismadefordatainthiscomputer.Forsimplicity,wecallsucharollbackandbackupprocessesisbackupoperationinthefollowingsections.ItisassumedthatcTandcRbetherespectivecostsforperiodicandrandomchecks.Inaddition,whenafaultoccursattimetbetweenkTand(k+1)TorSj+1,wecarryoutbackupoperationfromfaultpointtothelatestcheckingtimekT.ThisincursalosscostcD(t−kT)whichincludesallcostsforprocessingtimeandrollbackoperationfromttokT.While,whenafaultoccursattimetbetweenSjand(k+1)TorSj+1,thisincursalosscostcD(t−Sj).Theprobabilitythattheprocessperformsrollbacktoperiodiccheckis∑∞∫(k+1)T∑∞∫kTG(t−x)dG(j)(x)dF(t),(23)k=0kTj=00andtheprobabilitythattheprocessperformsrollbacktorandomcheckis∑∞∫(k+1)T∑∞∫tG(t−x)dG(j)(x)dF(t),(24)k=0kTj=0kTwhere(23)+(24)=1. 273September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookInspectionsforaComputerSystem259Therefore,thetotalexpectedcostuntilbackupoperationis∑∞∫(k+1)T{∑∞∫kTC(T)=[cTk+cRj+cD(t−kT)]k=0kTj=00}∑∞∫(k+1)T{∑∞∫t×G(t−x)dG(j)(x)dF(t)+[ckTk=0kTj=0kT}+cj+c(t−x)]G(t−x)dG(j)(x)dF(t)RD∑∞∫∞=cTF(kT)+cRM(t)dF(t)+cDµ0k=1∑∞∫(k+1)T∑∞∫kT−c(kT)G(t−x)dG(j)(x)dF(t)DkT0k=0j=0∑∞∫(k+1)T∑∞∫t+xG(t−x)dG(j)(x)dF(t).(25)kTkTk=0j=0Clearly,whenG(t)≡0,i.e.,thesystemischeckedonlyatperiodictimeskT(k=1,2,···),∑∞CP(T)=(cT−cDT)F(kT)+cDµ,(26)k=1whichagreeswith(5.55)of(Nakagawa,2008).WhenT=∞,i.e.,thesystemischeckedonlyatrandomtimesSj(j=1,2,···),∫∞∑∞∫tC(G)=cµ−xG(t−x)dG(j)(x)dF(t)RD00j=0∫∞+cRM(t)dF(t),(27)0whichagreeswith(Nakagawa,2008).WhenG(t)=1−e−µt,thetotalexpectedcostin(25)is∑∞µc∑∞∫(k+1)TC(T)=cF(kT)+c+D[1−e−µ(t−kT)]dF(t).TRλµkTk=1k=0(28)Inparticular,whenF(t)=1−e−λt,[]ccµcλ1−e−(µ+λ)TTRDC(T)=++1−.(29)eλT−1λµµ+λ1−e−λT 274September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book260ReliabilityModelingwithApplicationsClearly,C(0)≡limC(T)=∞,T→0cRµcDC(∞)≡limC(T)=+.T→∞λµ+λWefindanoptimalT∗(0 275September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookInspectionsforaComputerSystem261Anoptimumµ∗whichminimizesC(µ)iseasilygivenbyR√λλcR=,(35)µ+λcDandtheresultingcostis()C(µ∗)λµ∗R=+1.(36)cD/λµ∗+λµ∗+λClearly,ifλc/c≥1,then1/µ∗=∞,andc(0)=c/λ.RDRDTable4presentsT∗,1/µ∗andtheircostsC(T∗)/c,C(µ∗)/cforPDRDc/cwhenc=candλ=1.ThisindicatesthatT∗>1/µ∗whenc/cTDTRTDissmall,andC(T∗) 276September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book262ReliabilityModelingwithApplicationswhichdecreasesstrictlywithµbfrom1to0,andcompute()2bcR1=.cDµb+1Table5presents1/µb,bcR/cDandbcR/cT,andindicatesthatbcRisalittlelessthanthehalfofcT.Inotherwords,whentherandomcheckingcostisthehalfoftheperiodicone,twoexpectedcostsarealmostthesame.Forexample,C(T∗)/cwhenc/c=PTTD0.002,0.010,0.020,0.100,0.200,1.000,arealmostequaltoC(µ∗)/cwhenRTcT/cD=0.001,0.005,0.010,0.050,0.100,0.500,respectively,inTable4.Table5Valuesof1/µb,bcR/cDandbcR/cTwhenλ=1.cT/cD1/µbbcR/cDbcR/cT0.0010.0230.00050.50000.0020.0320.00100.50000.0050.0520.00250.50000.0100.0750.00490.49000.0200.1090.00970.48500.0500.1820.02370.47400.1000.2730.04600.46000.2000.4240.08870.44350.5000.8200.20300.40601.0001.5110.36230.36233.3NthRandomInspectionISupposethatthecomputersystemischeckedateveryNth(N=1,2,···)randomprocesstimes,i.e.,S,S,···.ByreplacingG(t)withG(N)(t)N2Nin(27)formally,thetotalexpectedcostuntilbackupoperationis∫∞C(N)=cM(N)(t)dF(t)+cµR1RD0∫∞∑∞∫t−cx[1−G(N)(t−x)]dG(jN)(x)dF(t),(37)D0j=00whereM(N)(t)≡∑∞G(jN)(t).j=1Inparticular,whenF(t)=1−e−λtandG(t)=1−e−µt,ANc(1−A)2∑NDjCR1(N)=cR1−AN+λA(1−AN)jA(N=1,2,···),(38)j=1 277September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookInspectionsforaComputerSystem263whereA≡µ/(µ+λ).FromtheinequalityCR1(N+1)−CR1(N)≥0,∑NjcR(1−A)(1−A)≥(N=1,2,···).(39)cD/λj=1Therefore,thereexistsafiniteanduniqueminimumN∗(1≤N∗<∞)11whichsatisfies(39).If[λ/(µ+λ)]2≥c/(c/λ)thenN∗=1.NotethatRD1N∗decreaseswithA,i.e.,N∗decreaseswith1/µfrom∞toaminimumN11integersuchthatN≥cR/(cD/λ).Next,weobtaintheexpectedcostuntilNprocesseshavebeencom-pleted.First,supposethatN=1.WhensystemfaultoccursbetweenSjandSj+1,wecarryoutbackupoperationtothelatestcheckingtimeSj.Then,theexpectedcostuntilcompletionofoneprocessisgivenbytherenewalequation∫∞∫∞CeR(1)=cRF(t)dG(t)+[cDt+CeR(1)]G(t)dF(t).(40)00TheexpectedcostuntilcompletionofNprocessesis,byreplacingG(t)withG(N)(t)formally,∫∞ct[1−G(N)(t)]dF(t)Ce(N)=c+D0∫(N=1,2,···).(41)RR∞G(N)(t)dF(t)0Asanapproximateobjectivefunction,weadopttheexpectedcostperoneprocessgivenby∫∞Ce(N)cct[1−G(N)(t)]dF(t)C(N)≡R=R+D0∫(N=1,2,···).R2∞NNNG(N)(t)dF(t)0(42)Inparticular,whenF(t)=1−e−λtandG(t)=1−e−µt,cc(1−A)2∑NRDjCR2(N)=+jA(N=1,2,···).(43)NλNAN+1j=1FromtheinequalityCR2(N+1)−CR2(N)≥0,∑N1−AjcR(1−A)≥(N=1,2,···),(44)AN+1cD/λj=1whoseleft-handsideincreasesstrictlyfrom(λ/µ)2to∞.Therefore,thereexistsafiniteanduniqueN∗(1≤N∗<∞)whichsatisfies(44).If22(λ/µ)2≥c/(c/λ),thenN∗=1,i.e.,weshouldplacecheckingtimesatRD2everycompletionofprocess.Clearly,N∗decreaseswith1/µfrom∞to1.2Comparedto(39)and(44),N∗≥N∗.12 278September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book264ReliabilityModelingwithApplications3.4NthRandomInspectionIIIthasbeenassumeduntilnowthatfaultisdetectedimmediately.Supposethatthefaultisdetectedonlyatrandomcheckingtimesandbackupoper-ationisexecuteduntilthelatestcheckingtime.Then,thetotalexpectedcostuntilbackupoperationis∑∞∫∞{∫∞}C(1)=cx[F(t+x)−F(t)]dG(x)dG(j)(t)MDj=000∑∞∫∞+cG(j)(t)dF(t).(45)Rj=00SupposethatthesystemischeckedateveryNth(N=1,2,···)randomtimes,i.e.,S,S,···.ByreplacingG(t)withG(N)(t)in(45),thetotalNN+1expectedcostuntilbackupoperationis∑∞∫∞{∫∞}C(N)=cx[F(t+x)−F(t)]dG(N)(x)dG(Nj)(t)M1Dj=000∑∞∫∞+cG(Nj)(t)dF(t)(N=1,2,···).(46)Rj=00Inparticular,whenF(t)=1−e−λtandG(t)=1−e−µt,∑∞(NN)∑∞C(N)=c(AN)j+c−AN(AN)jM1RDµµ+λj=0j=0ccN(1−A)(1−AN+1)RD=+(N=1,2,···).1−ANλA(1−AN)FromtheinequalityCM1(N+1)−CM1(N)≥0,[](1−A)2(1−AN)(1−AN+2)cR−N≥(N=1,2,···),(47)A(1−A)2ANcD/λwhosebracketontheleft-handincreasesstrictlywithNto∞.Clearly,theleft-handsideof(47)decreaseswithA,i.e.,N∗decreaseswith1/µfrom∞3to1.Therefore,thereexistsafiniteanduniqueN∗(1≤N∗<∞)which33satisfies(47).Next,supposethatwhensystemfaultoccursbetweenSjandSj+1,itsfaultisdetectedattimeSj+1andwecarryoutthebackupoperationfromSj+1toSj.Then,theexpectedcostbetweenoneprocessisgivenbytherenewalequation∫∞∫∞CeM(1)=cRF(t)dG(t)+[cDt+CeM(1)]F(t)dG(t).(48)00 279September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookInspectionsforaComputerSystem265TheexpectedcostuntilthecompletionofNprocessesis,byreplacingG(t)withG(N)(t),∫∞ctF(t)dG(N)(t)Ce(N)=c+D∫0(N=1,2,···).(49)MR∞F(t)dG(N)(t)0Thus,theexpectedcostperoneprocessis∫∞Ce(N)cctF(t)dG(N)(t)C(N)≡M=R+D∫0.(50)M2∞NNNF(t)dG(N)(t)0Inparticular,whenF(t)=1−e−λtandG(t)=1−e−µt,cc(1−A)(1−AN+1)RDCM2(N)=+(N=1,2,···).(51)NλAN+1FromtheinequalityCM2(N+1)−CM2(N)≥0,()21−AN(N+1)cR≥(N=1,2,···),(52)AANcD/λwhoseleft-handincreasesstrictlyto∞.Therefore,thereexistsafiniteanduniqueN∗(1≤N∗<∞)whichsatisfies(52).Ifλ(µ+λ)/µ3≥c/(2c/λ)44RDthenN∗=1.Clearly,theleft-handsideof(52)decreaseswithA,i.e.,N∗44decreaseswith1/µfrom∞to1,andN∗≥N∗.34Table6presentsoptimalN∗,N∗,N∗,andN∗whenc/(c/λ)=0.11234RDforλ/µ.ThisindicatesthatallN∗decreaseswithλ/µandN∗≥N∗andi12N∗≥N∗,andarealmostthesameforlargeλ/µ.34Table6OptimalN,N,N,andNwhencR/(cD/λ)=0.1.1234λ/µNNNN12340.01493932280.02252016140.05108660.1054330.2032220.501111 280September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book266ReliabilityModelingwithApplications4ConclusionsWehavefirstlyusedastandardinspectionpolicywithimperfectchecktodetectfaultinacomputersystem.Forthefirstmodel,thesystemischeckedatperiodictimesandrandomtimestodetectfaultsaccordingtocomputerprocesses,whicharecalledperiodicinspectionandrandominspectioninthischapter.Systemfaultisdetectedatthenextcheckingtimewithacertainprobabilityandundetectedfaultisdetectedatthesecondcheckingtimewiththesameprobability.Forthesecondmodel,wehaveappliedperiodicandrandominspectionsintofaultdetection,rollbackoperation,andbackupprocessforacomputersystem.Fortheseproposedtwomodels,expectedcostsofperiodicandrandominspectionshavebeenobtainedandtheoptimalinspectiontimeswhichminimizethemhavebeenderivedanalytically,whenthefailureandrandomtimesareexponential.Ithasbeenshownnumericallythatwhencostsforperiodicandrandomchecksarethesame,periodicinspectionsarebetterthanrandompolicies.However,itisofinterestthatifamodifiedcostforrandomcheckislowerthanthatforperiodicpolicy,randominspectionwouldbebetterandmorepractical.Furthermore,wehaveconsideredtheinspectionpolicyinwhichthesystemischeckedattheNthintervalandderivedanalyticallyoptimalNwhenthefaulttimeisexponential.Twocaseswhenthefaultisdetectedimmediatelyandatthefollowingcheckingtimehavebeendiscussedforthemodelthatisanapplicationofinspections.ReferencesBarlow,R.E.andProschan,F.(1965).MathematicalTheoryofReliability(Wiley,NewYork).Castillo,X.,McConner,S.R.andSiewiorek,D.P.(1982).Derivationandcalibra-tionofatransienterrorreliabilitymodel,IEEETransactionsonComputersC-31,pp.658–671.Fukumoto,S.,Kaio,N.andOsaki,S.(1992).Astudyofcheckpointgenera-tionsforadatabaserecoverymechanism,Computer&MathematicswithApplications24,pp.63–70.Liu,Y.M.andTraore,I.(2007).Propertiesforsecuritymeasuresofsoftwareproducts,AppliedMathematics&InformationSciences1,pp.129–156.Malaiya,Y.K.andSu,S.Y.H.(1981).Reliabilitymeasureofhardwareredun-dancyfault-tolerantdigitalsystemswithintermittentfaults,IEEETrans-actionsonComputersC-30,pp.600–604.Malaiya,Y.K.(1982).Linearlycorrectedintermittentfailures,IEEETransac-tionsonReliabilityR-31,pp.211–215. 281September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookInspectionsforaComputerSystem267Maeji,S.,Naruse,K.andNakagawa,T.(2010).Optimalcheckingmodelswithrandomworkingtimes.In:Chukova,S.,Haywood,J.andDohi,T.(Eds)AdvancedReliabilityModelingIV(McGraw-Hill,Taiwan),pp.488–495.Nakagawa,T.,Yasui,K.andSandoh,H.(1993).Anoptimalpolicyforadatatransmissionsystemwithintermittentfaults,IEICETransactionsonFun-damentalsofElectronics,CommunicationsandComputerSciencesJ76-A,pp.1201–1206.Nakagawa,T.(2005).MaintenanceTheoryofReliability(Springer,London).Nakagawa,T.(2008).AdvancedReliabilityModelsandMaintenancePolicies(Springer,London).Nakagawa,T.(2011).StochasticProcesswithApplicationstoReliabilityTheory(Springer,London).Nakamura,S.andNakagawa,T.(2010).StochasticReliabilityModeling,Opti-mizationandApplications(WorldScientific,Singapore).Naruse,K.,Nakagawa,T.andMaeji,S.(2009).RandomcheckpointmodelswithNtanemtasks,IEICETransactionsonFundamentalsofElectronicsCom-municationsandComputerSciencesE92-A,pp.1572–1577.Qian,W.,Yin,X.andXie,L.(2012).SystemreliabilityallocationbasedonBayesiannetwork,AppliedMathematics&InformationSciences6,pp.681–687.Reuter,A.(1984).Performanceanalysisofrecoverytechniques,ACMTransactiononDatabaseSystems9,pp.526–559.Su,S.Y.H.andKoren,T.andMalaiya,Y.K.(1978).AContinuous-paremetermarkovmodelanddetectionproceduresforintermittentfaults,IEEETransactionsonComputersC-27,pp.567–570.Savsar,M.andAldaihani,M.(2012).Astochasticmodelforanalysisofmanufacturingmodules,AppliedMathematics&InformationSciences6,pp.587–600.Xiao,Y.,Zhang,H.,Xu,G.andWang,J.(2012).Apredictionrecoverymethodforsupportingreal-timedataservices,AppliedMathematics&InformationSciences6-2S,363S–369S.Yasui,K.,Nakagawa,T.andSandoh,H.(2002).Reliabilitymodelsinreliabilityandmaintenance.In:S.Osaki(Ed.)StochasticModelsinReliabilityandMaintenance(Springer,Berlin),pp.281–301.Zhao,X.andNakagawa,T.(2012).Optimizationproblemsofreplacementfirstorlastinreliabilitytheory,EuropeanJournalofOperationalResearch223,pp.141–149.Zhang,Y.andXiao,J.(2012).Asoftwarereliabilitymodelingmethodbasedongeneexpressionprogramming,AppliedMathematics&InformationSciences6,pp.125–132. 282May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank 283September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookPART4ReliabilityApplications269 284May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank 285September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookChapter15DynamicFaultTreeAnalysisTetsushiYugeandShigeruYanagiDepartmentofElectricalandElectronicEngineering,NationalDefenseAcademy,Hashirimizu1-10-20,Yokosuka239-8686,Japan1IntroductionFaulttreeanalysis(FTA)isoneoftheoldest,mostimportantlogicandprobabilistictechniquesinindustrialapplications[Leeetal.(1985);Vesely(1981);Stamatelatos(2002)].Itcanbesimplydescribedasananalyticaltechnique,wherebyanundesiredstateofasystem,calledatopevent(TE),isspecified(astatethatiscriticalfromasafetyorreliabilitystandpoint),andthesystemisthenanalyzedinthecontextofitsenvironmentandoperationtofindallrealisticwaysinwhichtheTEcanoccur.Thefaultscanbeeventsthatareassociatedwithcomponenthardwarefailures,humanerrors,softwareerrors,oranyotherpertinenteventswhichcanleadtotheundesiredevent.Afaulttree(FT)thusdepictsthelogicalinterrelationshipsofbasiceventsthatleadtotheTE.OneofthemainaimsinFTAistoobtaintheexactTEprobability,buttheexactanalysisofareasonablylarge-scaleFTwithacomplexstructure,suchasthatforachemicalplant,anuclearreactororanairplane,canbeexpensive,intermsofboththetimerequiredtodeveloptheFTmodelandthetimerequiredtosolvethemodel.Severaltypesofdynamicbehaviorinsuchasystemaswellasthescaleofthesystemmaketheanalysisdifficult.Examplesofdynamicbehaviorincludetransientrecovery,intermittenter-rors,andsequencedependence[Duganetal.(1993)].Dynamicfaulttree(DFT)analysis,thatisanextensionoftraditionalstaticFT(SFT)anal-ysis,allowsthemodelingofdynamicbehavior.DFTstakeintoaccountnotonlythecombinationoffailureeventsbutalsotheorderinwhichthey271 286December19,201312:19BC:9023-ReliabilityModelingwithApplications2013book272ReliabilityModelingwithApplicationsoccur.AsthislastaspectisnottakenintoaccountintheBooleanmodeloffailures(whichonlyexpresseswhetherabasiceventhasoccurredornot),aclassicalBooleanfunctioncannotrepresentthedynamicrelationsbetweentheTEandthebasiceventsthatexistinaDFT.TheDFThasbeencontinuouslystudiedinthelasttwodecades.Wereviewtherecentextensionsoftheanalysisinthischapter.InSec.2,abriefintroductionondynamicgatesusedinaDFTispresented.TherepresentativetechniquestosolveDFTintermsofquantitativeanalysisareintroducedinSec.3.TherecentextensionofthealgebraicapproachintermsofbothqualitativeandquantitativeevaluationsispresentedinSec.4.2DynamicGatesDFTdefinesspecialgatesthatcaptureavarietyoffailuresequencesandfunctionaldependences.Sixdynamicgatesareproposedin[Duganetal.(1993)]:priorityAND(PAND)gate,functionaldependency(FDEP)gate,hotspare(HSP)gate,warmspare(WSP)gate,coldspare(CSP)gate,andsequenceenforcing(SEQ)gate.Fig.1Dynamicgates:(a)PANDgate,(b)FDEPgate,(c)WSPgateand(d)SEQgate.Thefirstdynamicgate,PANDgate,wasintroducedin1976tomodelsequencesoffailures[Fusselletal.(1976)].ItislogicallyequivalenttoanANDgate,ofwhichinputeventsmustoccurinaspecificorderfortheoutputeventtooccur.TheoutputofaPANDgatewithtwoinputsdepictedinFig.1(a)becomestrueifandonlyifbothbasiceventsAandBhaveoccurredandbasiceventAhasoccurredbeforebasiceventB.Asaseconddynamicgate,FDEPgatewasintroducedin1990tomodelcommoncausefailures[Duganetal.(1990)].TheFDEPgateinFig.1(b) 287September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookDynamicFaultTreeAnalysis273hasasingletriggerinput,i.e.,eventA,anon-dependentoutput(reflectingthestatusofthetriggerevent)andoneormoredependentbasicevents.Thedependenteventsarefunctionallydependentonthetriggerevent.Whenthetriggereventoccurs,dependenteventsBandCareforcedtooccur.AnFDEPgatecanbeusedtoreflectsuchdependenceexplicitly.Theoccurrenceofanyofthedependenteventshasnoeffectonthetriggerevent.Asaderivativegate,theprobabilisticdependent(PDEP)gateispro-posed[Montanietal.(2005)].Thetriggereventcausesotherdependenteventswithprobabilitypdep≤1inthePDEPgate.Whenpdep=1,thegateconsistswiththeFDEPgate.TheCSPandHSPgateswerealsointroducedin[Duganetal.(1990)].TheWSPgategeneralizestheCSPandHSPgates,andintroducedin1998[RenandDugan(1998)].Figure1(c)showsaWSPgatewithtwowarmspares.BasiceventAcorrespondstoaprimaryunitwhichisoriginallypoweredon,andtheotherinputsspecifythecomponentsusedasreplace-mentsfortheprimaryunit.Thefailureratesofspareunitsarereducedbyafactorα,calledthedormancyfactorinstandbymode.Ifα=0,thegateisaCSPgate.Ifα=1,itisregardedasanHSPgate.TheWSPgatehasoneoutputthatbecomestrueafteralltheinputeventsoccur.SEQgatewasintroducedin1993[Duganetal.(1993)].TheSEQgateinFig.1(d)forceseventstooccurinaparticularorder.Theinputeventsareconstrainedtooccurintheleft-to-rightorder,i.e.,theleftmosteventmustoccurbeforetheeventonitsimmediaterightwhichmustoccurbeforetheeventonitsimmediaterightisallowedtooccur,etc.ManyauthorshavedescribedtheactualexamplesofDFT.Table1givestheinformationoftheexamplesandhelpsustounderstandDFTandse-quencedependency.3AnalysisofDynamicFaultTreeSeveralapproacheshavebeenusedtoanalyzeDFTs.Theseapproachescanbeeithermodularorglobal[Merleetal.(2011b)].GlobalapproachesconsistinsolvingthewholeDFTdirectly,whereasmodularapproachesconsistin:•DividingtheDFTintoindependentstaticanddynamicsubtrees(ormodules)priortoanalysis.Thelineartimealgorithm[DutuitandRauzy(1996)]isausefulmethodtosearchmodulesofFT,wheredynamicgates 288September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book274ReliabilityModelingwithApplicationsTable1ExamplesofDFT.FaultTolerantParallelProcessor[Duganetal.(1990)],[Duganetal.(1992)],[Duganetal.(1993)]MissionAvionicSystem[TangandDugan(1994)],[Dugan(2000)]ActiveHeatRejectionSystem[BoudaliandDugan(2005a)],[Montanietal.(2008)]MultiprocessorComputingSystem[Stamatelatos(2002)],[Montanietal.(2006)],[Zhangetal.(2009)],[Merleetal.(2011a)]Redundantsystemwithswitch[Fusselletal.(1974)],[Stamatelatos(2002)],[YugeandYanagi(2008)],[Merleetal.(2010)]HypotheticalCardiacAssistSystem[RenandDugan(1998)],[BoudaliandDugan(2005b)],[Montanietal.(2005)],[Merleetal.(2011b)]VehicleManagementSystem[Stamatelatos(2002)]canbetreatedinthesamewayasstaticgates.Ifasubtreecontainsstaticgatesonly,itisconsideredasstatic,whereas,asubtreecontainsatleastonedynamicgate,itisconsideredasdynamic.•Solvingthemodulesseparately.•Combiningtheresultsofthevariousmodulestoobtaintheoverallresultfortheentiretree.Generally,modularapproacheshavebettercalculationefficiencythanglobalapproaches.Inmodularapproach,solvingstaticmodulescanbedonebyusingBinaryDecisionDiagrams(BDDs)orothercombinatorialtechniques.Ontheotherhand,themethodsofsolvingdynamicmodulesarerestrictivebecauseaclassicalBooleanfunctioncannotrepresentthedy-namicbehaviors.Followingsaretheproposedmethodstoanalyzedynamicmodules.Table2classifiesthereferences.3.1MarkovAnalysisMarkovchain(MC)isawell-knowntooltocapturethebehaviorsofasys-temwhichcontainvarioustypesofdependencesbetweenbasicevents.MCsandtheirextensionshavebeenprovedtobeaversatiletoolformodelingDFTs[Duganetal.(1992,1993);Dugan(2000);OuandDugan(2004);GulatiandDugan(1997);RenandDugan(1998)].Thesecanbeappliedtobothentiretreesandsubtrees.However,theMC-basedapproachesarebesetwithtwowell-knownproblems: 289September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookDynamicFaultTreeAnalysis275Table2Classificationofreferences.Staticfaulttreesurvey[Leeetal.(1985)],[Vesely(1981)],[Stamatelatos(2002)]minimalcutset[Fusselletal.(1974)]modularization[DutuitandRauzy(1996)]Bayesiannetwork[Pearl(2000)],[Bobbioetal.(2001)],[WeberandJouffe(2003)]sumofdisjointproducts[Abraham(1979)],[Locks(1987)]DynamicfaulttreeMarkovanalysis[Fusselletal.(1976)],[Duganetal.(1992)],[Duganetal.(1993)],[GulatiandDugan(1997)],[RenandDugan(1998)],[Dugan(2000)],[OuandDugan(2004)]Algebraicapproach[Fusselletal.(1976)],[TangandDugan(1994)],[YugeandYanagi(2007)],[YugeandYanagi(2008)],[Yonedaetal.(2010)],[Merleetal.(2010)],[Merleetal.(2011a)],[Merleetal.(2011b)]Bayesiannetwork[WeberandJouffe(2003)],[Montanietal.(2005)],[BoudaliandDugan(2005a)],[BoudaliandDugan(2005b)],[BoudaliandDugan(2006)],[Montanietal.(2006)],[Montanietal.(2008)],[YugeandYanagi(2012)]MonteCarlo&Petri-net[Longetal.(2002)],[BobbioandCodetta-Raiteri(2004)],[Zhangetal.(2009)]Repairableevents[Duganetal.(1990)],[Duganetal.(1993)],[BobbioandCodetta-Raiteri(2004)],[Yugeetal.(2012)](1)Ineffectivenessinsolvinglargedynamicmodules,i.e.thenumberofstatesgrowsexponentiallyasthenumberofbasiceventsinthemoduleincreases.(2)Lackofmodelingpowercapabilities,i.e.,thefailuretimedistributionofabasiceventislimitedtotheexponentialdistribution.Concerningtheproblem(1),theexistenceofarepeatedeventcausesalargemodule.Arepeatedeventisdefinedasaneventconnectedtoseveralgates.Ifoneoftheinputeventsofadynamicgateissharedbyanothergate,wehavetotacklealargemodule.Inadynamicmodule,thedimensionalityofMCreachestothetotalnumberofbasiceventsinthemoduleforthesakeofcapturingeventsequencesprecisely.Therefore,theconstructionandanalysisofanMCaretediousandrequirehugecomputationalresourcesasthenumberofbasiceventsinamoduleincreases(seeTable3).Evenifthemodularizationisadopted,inmanycases,thesizeofasinglemoduleremainssignificant,potentiallyleadingtoanunreasonablylongcomputa- 290September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book276ReliabilityModelingwithApplicationsTable3Numberofstatesindynamicmodule.numberofeventsnumberofstatesinadynamicmoduletobeconsideredinMC25465619,5578109,60110986,410153.55×1012tiontime.Hence,theapplicabilityofthemodularizationtechniqueisnotalwaysevident.Theproblem(2)isintrinsicinMC,wehavetotryanothermethodtoavoidtheproblem.3.2AlgebraicApproachAsamethodofextendingthemodelingcapability,i.e.,asanalternativemethodtodealwithproblem(2)inMC,thealgebraicapproachhasbeenproposed.Merleetal.proposedacompletealgebraicexpressionofDFTswithprioritydynamicgates[Merleetal.(2010)].PrioritydynamicgatesindicatethePANDandFDEPgates,whichexpresssemanticsofpriority.Theyintroducednewtemporaloperatorstodefinethesequencedependenceofsuchgates.ByusingtheoperatorsandconventionalBooleanoperators,theoccurrencetimeofaTEisrepresentedasasumoftheproductcanonicalform.ThetermsofthisformcorrespondtotheminimalcutsequencesetoftheDFT.TheTEprobabilityiscalculatedfromthecutsequencesetbyaninclusion-exclusiontechnique.Thismethoddoesnotrequirearestrictiononthefailuretimedistribution.However,toobtaintheprobabilityofacutsequence,oneneedstosolvemultipleintegrations1.Thismayrestrictthesequencesizetohandleintheanalysis.Furthermore,asthenumberofminimalcutsequencesetsinaDFTtendstobelargecomparedwiththatinastaticFT,evenifthescales(numberofbasiceventsorgates)oftheFTsarealmostthesame,theapplicationoftheirinclusion-exclusion-basedmethodtothecalculationofaTEprobabilityislimitedtoverysmallFTs.TheycontinuouslystudiedalgebraicexpressionsforDFTswithsparegates[Merleetal.(2011a,b)].ThedetailsofthisapproachandtherelatedstudiesarediscussedinSec.4.1Thecomplexityofthecalculationmayrestricttheusageofthefailuredistributionexceptinthecaseofperformingthenumericalcalculation. 291September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookDynamicFaultTreeAnalysis2773.3BayesianNetworkTheuseofaBayesiannetwork(BN)[Pearl(2000)]inDFTanalysisisexpectedtobeaneffectivemethodtoovercomebothproblemsinMC.TheBNdirectlytakesintoaccounttherepeatedeventswithoutaspecificcomputation.ThismakestheBNefficientforobtainingexactresults.AnSFTcanbetransformedtoanequivalentBNwithoutlosingtheone-to-onematchtotheactualstructureofthesystem[Bobbioetal.(2001)].In[BoudaliandDugan(2005b)],BoudaliandDuganproposedaninterval-baseddiscrete-timeBN(DTBN)forthereliabilityanalysisofDFTs.Theyintroducedmissiontimeanddivideditintonintervals.ThevariableassignedtoanodeintheDTBNhasn+1states.Ther.v.Xisinstatexifandonlyifthenodehasoccurredinthex-thtimeinterval.Thelaststaterepresentsthesurvival(nooccurrence)forthedurationofmissiontime.ArcsthatconnectpairsofnodesintheDTBNrepresentthecausalprobabilisticrelationshipbetweennodes.Thesecausalprobabilitiesarespecifiedbydefiningn+1dimensionalconditionalprobabilitytables(CPTs).TheDTBNisgeneratedusingastandardBNinferencealgorithm.However,thisisanapproximatesolutionandrequireshugememoryre-sourcestoobtainthejointprobabilitydistribution(JPD)accurately.BoudaliandDugansequentiallyproposedacontinuous-timeBN(CTBN)forthereliabilityanalysisofDFTs[BoudaliandDugan(2006)].Ifthenumberofstates,n,tendstoinfinity,theDTBNbecomesaCTBN.TheCTBNgivesanexactclosed-formsolutionanddoesnotrequiretheCPTofanode.Themarginalprobabilitydistribution(MPD)ofanr.v.representingagateofaDFTisexpressedbythemultipleintegralformus-ingthefailuredistributionsofinputevents.InaCTBN,theCPTscannotbeusedforevaluatingtheconditionalprobabilitiesofnodes.ThisrestrictsthehierarchicalfeaturesofBNandmakestheanalysiscomplex,becauseonehastoderivealltheconditionalprobabilitiesanalytically.Montanietal.[Montanietal.(2008)]proposedatranslationoftheDFTintoadynamicBayesiannetwork(DBN).Theirmethodwasonthebasisofthe2-time-sliceDBN[Montanietal.(2006);WeberandJouffe(2003)].The2-time-sliceDBNisequivalenttoanMC,i.e.,theybothpos-sesstheMarkovproperty.ThustheDBNmodelisapplicableexclusivelytoMarkovprocesses.Furthermore,itshouldbementionedthatDBNanalysisisclassifiedasasimulationmethod,thus,theresultofthecalculationgivestheapproximatedprobability. 292September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book278ReliabilityModelingwithApplications3.4MonteCarloandPetri-netMonteCarloapproaches[Longetal.(2002)]arealsoavailableforsolvingDFTsthatdonotrequiretheassumptionofanexponentialdistributionforthefailuretimes.StochasticPetrinet(PN)isalsoappliedtoDFT[BobbioandCodetta-Raiteri(2004);Zhangetal.(2009)].BytransformingaDFTmodelintoaPNmodel,wecanavoidthebothoftheseriousproblemsinMarkovanalysis.However,modelingwithMonteCarlo/PNisverycomplexandinordertoachievehighprecision,itisnecessarytoincreasethenumberofcycles,whichresultinlongercomputationtime.Nevertheless,thesemethodsareusefultoderivetheprobabilisticresultsinacomplicatedDFTanalysis.3.5RepairableDynamicFaultTreeTheconventionalFTusuallyconsidersonlyfailureoccurrenceasaninputevent.However,consideringrepairimprovestheanalysiscapabilityofFT.ThisFTisreferredtoasarecoverable/repairableFT(RFT).MCissuitabletodealwithRFTevenifdynamicgatesareincludedin.Actually,somestudiesbyusingMCdealtwithRFT[Duganetal.(1990,1993)].Thesimulationbasedmethodsarealsosuitabletodealwithrepairevent[BobbioandCodetta-Raiteri(2004)].Reference[Yugeetal.(2012)]triedtohandlethestatetransitionsinMCasequivalenteventoccurrencesbyusingtheconceptofrenewalprocess.Thatis,theoccurrenceandrestorationofagateoutputwereregardedasfollowinganalternativerenewalprocess.Then,thesteadystateprobabilityofanFTwasderivedfromthelimittheoremofarenewalprocess.4AlgebraicExpressionofDFT4.1TemporalFunctionsandOperatorsTheBooleanmodel,whichrealizesthecompletealgebraicexpressionofSFT,cannotrendertheorderofoccurrenceofeventsdefinedbydynamicgates.Inordertotakeintoaccountthepriorityrelations,Merleetal.con-siderthetemporalfunctionsandtemporaloperators[Merleetal.(2010)].Allthebasiceventsandtheintermediateeventsaredefinedastemporalfunctionswhicharepiecewiseright-continuousonR+∪{+∞},andwhoserangeareB={0,1}.Theuniquedateofoccurrenceofeventaisdefined 293September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookDynamicFaultTreeAnalysis279asd(a).Theidentityelementsequivalentto0and1aredenotedby⊥and⊤towhichthesedatescanbeassignedas;d(⊥)=+∞,d(⊤)=0.Theyalsointroducedthreeoperators,BEFORE(▹),SIMULTANEOUS(△)andINCLUSIVEBEFORE(E),basedonoccurrenceofaandb,asfollows;aifd(a) 294September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book280ReliabilityModelingwithApplications4.2MinimalCutSequencesSet4.2.1DFTwithPANDgatesLetusconsiderthealgebraicexpressionsofdynamicgatesbyusingthetemporalfunctionsandoperators.TheoutputofthePANDgateinFig.1(a),sayQ,isgivenin[Merleetal.(2010)]asQ=B(A▹B).Notethattheomissionofalogicalconjunctionsymbol(·)isadmittedhereafter.ForaPANDgatehavinginputse1,e2,...,enandaconditionthattheorderoftheinputeventsisspecifiedinthisorder,theoutputQisexpressedasfollows,Q=en(e1▹e2)(e2▹e3)···(en−1▹en).(14)LetusconsidertheFTinFig.2.Theoutputofthetopeventisex-pressed,usingtherelationsinEq.(1)–Eq.(14)),asfollows,TE=G2(G1▹G2)=(e3+e4)(G1▹G2)=e3(G1▹G2)+e4(G1▹G2)=e3((G3(e1▹G3))▹G2)+e4((G3(e1▹G3))▹G2)=e3(G3▹G2)((e1▹G3)▹G2)+e4(G3▹G2)((e1▹G3)▹G2)=e3(G3▹G2)(e1▹G3)(e1▹G2)+e4(G3▹G2)(e1▹G3)(e1▹G2)=e3((e2+e3)▹G2)(e1▹G3)(e1▹G2)+e4((e2+e3)▹G2)(e1▹G3)(e1▹G2)=e3(e2▹G2)(e1▹G3)(e1▹G2)+e3(e3▹G2)(e1▹G3)(e1▹G2)+e4(e2▹G2)(e1▹G3)(e1▹G2)+e4(e3▹G2)(e1▹G3)(e1▹G2)=e3(e2▹e3)(e2▹e4)(e1▹e2)(e1▹e3)(e1▹e3)(e1▹e4)+e3(e3▹e3)(e3▹e4)(e1▹e2)(e1▹e3)(e1▹e3)(e1▹e4)+e4(e2▹e3)(e2▹e4)(e1▹e2)(e1▹e3)(e1▹e3)(e1▹e4)+e4(e3▹e3)(e3▹e4)(e1▹e2)(e1▹e3)(e1▹e3)(e1▹e4)=e3(e2▹e3)(e2▹e4)(e1▹e2)+e3(e3▹e4)(e1▹e2)(e1▹e3)+e4(e2▹e3)(e2▹e4)(e1▹e2)+e4e3(e3▹e4)(e1▹e2)(e1▹e3)=e3(e2▹e3)(e2▹e4)(e1▹e2)+(e3▹e4)(e1▹e2)(e1▹e3)+e4(e2▹e3)(e2▹e4)(e1▹e2)+e4(e3▹e4)(e1▹e2)(e1▹e3)(15)Finally,fourproducttermsarederivedfromtheDFT.Equation(15)isthesumofcanonicalformoftheDFT.InthecaseofstaticFTs,suchformprovidesthecutsetsofanFT.InthecaseofDFTs,theconceptofcutsetcorrespondstocutsequencerepresentingtheorderedfailuresequenceofeventsthatcausestheTEtooccur.Therefore,Eq.(15)showsthecut 295December19,201312:19BC:9023-ReliabilityModelingwithApplications2013bookDynamicFaultTreeAnalysis281Fig.2DFTwithtwoPANDgates.sequenceset(CSS)oftheDFT.SincethefourthCSSinEq.(15)isincludedinthesecondCSS,itcanberemoved.Ingeneral,CSSiisincludedinotherCSSandremovedifthefollowingequationholds,XCSSi·CSSj=CSSi.(16)j6=iAsaresultofthisminimization,theminimalcanonicalformoftheTEisexpressedasfollows,TE=e3(e2⊳e3)(e2⊳e4)(e1⊳e2)+(e3⊳e4)(e1⊳e2)(e1⊳e3)+e4(e2⊳e3)(e2⊳e4)(e1⊳e2).(17)Thisformalsoofferstheminimalcutsequenceset(MCSS)oftheDFTasfollows,MCSS={MCSS1,MCSS2,MCSS3}={e3(e2⊳e3)(e2⊳e4)(e1⊳e2),(e3⊳e4)(e1⊳e2)(e1⊳e3),e4(e2⊳e3)(e2⊳e4)(e1⊳e2)}.(18)NotethateachMCSSinEq.(18)isnotasinglecutsequence,butanalgebraicexpressionthatmaycontainmorethanonecutsequenceprovidingasufficientconditionontheorderofbasiceventfailuresthatleadtothetopevent.Forinstance,MCSS1inEq.(18)containsthreecutsequences,e¯4[e1,e2,e3],[e1,e2,e3,e4]and[e1,e2,e4,e3],here,[...]meanstheorderedcutsequence. 296December19,201312:19BC:9023-ReliabilityModelingwithApplications2013book282ReliabilityModelingwithApplications4.2.2DFTwithWSPgatesTheoutputofaWSPgateinFig.1(c),sayQ,willoccurifA,B,andCfailaccordingtosequences[A,B,C],[A,C,B],[B,A,C],[B,C,A],[C,A,B],or[C,B,A][Merleetal.(2011a)].ItisimportanttonotethatBandCwillnothavethesamedistributionfunctioninthesixsequences.Forinstance,insequence[A,B,C],bothBandCfailduringtheiractivemode(denotedbyBaandCa),whereasinsequence[B,C,A],bothBandCfailduringtheirdormantmode(denotedbyBdandCd).ThealgebraicmodeloftheWSPgatecanhencebeexpressedasfollows[Merleetal.(2011a)],Q=Ca(A⊳Ba)(Ba⊳Ca)+Ba(A⊳Cd)(Cd⊳Ba)+Ca(Bd⊳A)(A⊳Ca)+A(Bd⊳Cd)(Cd⊳A)+Ba(Cd⊳A)(A⊳Ba)+A(Cd⊳Bd)(Bd⊳A).(19)AsBandCcannotbebothinanactivestateandinadormantstate,wehaveBdBa=CdCa=⊥.(20)Equation(19)impliestheminimalcutsequencesoftheWSPgate.Thenumberofminimalcutsequencesisafactorialofthenumberofinputsevents.Next,letusconsideraDFTinFig.3.TheintermediateeventsandTEareexpressedusingthetheoremspresentedintheprevioussectionasFig.3ExampleofDFT. 297September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookDynamicFaultTreeAnalysis283followsG1=Ba(A▹Ba)+A(Bd▹A)+A(C▹A)G3=Ba(C▹Ba)+C(Bd▹C)+C(A▹C)G2=D(G3▹D)=(C▹Ba)(Ba▹D)+(Bd▹C)(C▹D)+(A▹C)(C▹D)TE=G0=G1+G2=Ba(A▹Ba)+A(Bd▹A)+A(C▹A)+D(C▹Ba)(Ba▹D)+D(Bd▹C)(C▹D)+D(A▹C)(C▹D)(21)Here,thelasttermofG1isaddedtorepresenttheconditionofsharedevent.ThespareeventBcannotbeinusewheneventAfails,ifeventChasalreadyfailed.ThelasttermofG3isaddedforthesamereason.Finally,TEisrepresentedasthesumofsixproductterms.EachofwhichrepresentstheCSSoftheDFT.Afterremovingredundantcutse-quencessetbyEq.(16),theMCSSaregenerated.AllsixtermsinEq.(21)areMCSSs.4.2.3FEDPandSEQgatesAsalreadynoticedin[Merleetal.(2010)],thealgebraicformalizationprovesthattheFDEPgatecanberepresentedbyBooleanORgates.ThatiswecanrepresentbasiceventBandCinFig.1(b)asvariablesB′=A+BandC′=A+C.TheSEQgatecanbeexpressedasaspecificcaseofWSPgate,i.e.,α=0.AnyexpressioncontainingBdandCdinEq.(19)canberemoved.4.3SequenceProbabilityAsafirstaccomplishmentconcernedwiththeprobabilisticanalysisofeventsequences,Fusselletal.[Fusselletal.(1976)]formulatedtheoccurrenceprobabilityofPANDgatein1976.ThesequenceprobabilityofninputsPANDgate,[e1,e2,...,en],isobtainedbyPr([e1,e2,...,en])(t)∫t∫xn∫xn−1∫x2=fn(xn)fn−1(xn−1)···f1(x1)dx1dx2...dxn,(22)0000where,xiindicatestheoccurrencetimeofeiandfi(t)isafailuretimedistri-butionofeventei.Theyassumedanexponentialdistributionwithparam- 298September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book284ReliabilityModelingwithApplicationseterλiforfi(t)andderivedthefollowingequation[Fusselletal.(1976)].∏n∑ne−aktPr([e1,e2,...,en])(t)=λin(23)∏i=1k=0(a−a)jkj=0,j̸=kwhere,∑na0=0,ai=λkfori>0.(24)k=iYugeandYanagi[YugeandYanagi(2012)]discussedaprobabilityofeventsequenceincludingseveralspareevents.Letusconsideraneventsequencecomposedwithnevents,e1,e2,...,en.Aneventinthesequenceisdenotedbyei,whichmeansthattheeventthatfailedinthej-thorderofjthesequenceisdesignatedaspareofaneventthatfailedinthei-thorder.e0denotesaneventthatwasoriginallyinactivemode.ei,(i̸=0)hasajjdormancyfactor0≤αj≤1.Onthebasisofthenotation,thesequenceisrepresentedas[ei1,ei2,...,ein].Theeventsinthesequenceareclassified12nintothefollowingsets[YugeandYanagi(2012)]:•S:Eventsthatwereoriginallyinactivemode,i.e.,e0.TheeventsinajSaaredividedintothefollowingtwosets:–Sar:Eventsinwhichthefunctionistakenoverbyanotheroneafterthefailure.–San:Eventsthathavenospareorthatcannotusetheirsparebecauseofthefailureofthedesignatedspare.•Ss:Eventsthatwereoriginallyinstandbymode.TheeventsinSsaredividedintothefollowingthreesets:–Sss:Eventsthatfailinstandbymodebeforethereplacement.Forei∈S,i>j.jss–Ssar:Owingtothefailureofthedesignatedevent,theoperationmodechangestoactivemodethenfailsinactivemode.Afterthefailure,thefunctioniscontinuouslytakenoverbyanotherspare.–Ssan:Eventsthathavenoinheritablespareaftertheirfailureinactivemode.LetS={S,S}.Forei∈S,i 299September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookDynamicFaultTreeAnalysis285Thesequenceprobabilityof[ei1,ei2,...,ein]canbederivedusingthe12nn-tupleintegrationas∫t∫xn∫x2∏∏Pr([ei1,ei2,...,ein])(t)=···f(x)f(x)12njjjαj000e0∈Sei∈Sjajss∏×F¯jα(xi)fj(xj−xi)dx1dx2...dxn,(25)ei∈Ssajwhere,f(x)isthefailuredensityfunctionofeiandF¯(x)isthesurvivaljjjαfunctionofeiinstandbymode.jEquation(25)isaclosedformsolutionapplicabletogeneralfailuredistributions.However,toobtainthesolution,oneneedstogothroughaseriesofsymbolicintegrations.Thisprocessistime-consumingandfeasibleonlyforrelativelysmallsequences.Ifthefailuredistributionsareassumedtobeexponential,thefollowingtheoremcanbeapplied[YugeandYanagi(2012)].Theorem15.1.Let[ei1,ei2,...,ein]beaneventsequencecontainingspare12neventsei∈{S,S}withthedormancyfactor,α,0≤α≤1.Whenthejsassjjfailuretimeofeiinactivemodefollowsanexponentialdistributionwithajmean1/λj,thesequenceprobabilityis{n()}i1i2in∏−11∏λiPr([e1,e2,...,en])(t)=αj·L(26)ss+aiei∈Sssi=1jwhere,∑n∑n∑nai=λk−(1−αk)λk−(1−αj)λjfori>0,(27)k=ik=ik=iek∈Sssek∈SsajandL−1istheinverseLaplacetransformoperator.Ifaneventsequenceisspecified,wecantransformtheinverseLaplaceexpressioninEq.(26).Withoutthespecification,itisnoteasytoobtainthesequenceprobabilitywithatimedomain.Asaspecialcase,ifeveryaiinEq.(27)isdistinctfromtheothers,thesequenceprobabilityis∏∏n∑ne−aktPr([ei1,ei2,...,ein])(t)=αλ,(28)12nji∏nei∈Sssi=1k=0j(aj−ak)j=0,j̸=kwhereaj,j>0,aregivenbyEq.(27)anda0=0. 300September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book286ReliabilityModelingwithApplicationsIfei,2≤i≤n,worksasacoldspareofei−1inthesequence,i.e.,[e0,e1,...,en−1],Eq.(26)isinagreementwiththeoccurrenceprobability12nofanninputsCSPgate.Furthermore,ini.i.d.case(λ1=...=λn=λ),theoccurrenceprobabilityis{}λt(λt)n−1Pr([e0,e1,...,en−1])(t)=1−e−λt1++...+.(29)12n1!(n−1)!ThisisanErlangdistributionwithphasenandparameterλ.Whenallthespareeventsinasequencearehotspareevents,these-quenceprobabilityisinagreementwiththeoccurrenceprobabilityofanninputsPANDgate,whoseoccurrenceconditionis[e1,e2,...,en],presentedinEq.(23).4.4TopEventProbabilityofDFTInthecaseofstaticFTs,awidespreadwaytocalculatetopeventprobabil-itiesistofollowaninclusion-exclusion(IE)ruleifalltheminimalcutsetsaregiven.ThistechniquecanbeappliedtoDFTs[Merleetal.(2010)].However,thenumberofMCSSsinDFTtendstobeenormouscomparedwiththatofastaticFTevenifthescaleoftheFTsalmostthesame.Fur-thermore,thecalculationofproducttermsgeneratedfromthetechniqueismorecomplexthanstaticFTcases.Therefore,applyinganIEbasedtechniqueislimitedtosmallsizeDFTs.Anotherpopularmethodforobtainingthereliabilityofacoherentsys-temismakinguseofthesumofdisjointproducts(SDP)algorithm[Abra-ham(1979)].Thealgorithmwasproposedinordertoanalyzenetworkreliability.ItstartswiththeBooleansumofproductscorrespondingtoeachofthesimplepathsbetweenthepairofnodes,thentransformsthisintosumofdisjointproducts,fromwhichthereliabilityexpressioncanbedirectlyobtained.ThismethodcanbeappliedtoobtainingthetopeventprobabilityofstaticFTswhentheminimalcutsetsaregiven.Inthiscase,thealgorithmbasesonthefollowingtheorem[Abraham(1979)].Theorem15.2.(Abraham’stheorem)LetSjbeaBooleanproductcorrespondingtoaminimalcutsetandPibeanyproductwhichimpliessomeminimalcutset.a)IfthereisatleastonevariablewhichexistsinSjandcomplementedinPi,thenSjandPiaredisjoint.b)IfSjandPiarenotdisjoint,letX≡{xa,xb,...,xc}bethesetofvariableswhichexistinSjandwhichdonotexistinPi.Then, 301September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookDynamicFaultTreeAnalysis287i)IfX=ϕ,thenSj∪Pi=Sj.ii)IfX̸=ϕthenSj∪Pi=Sj∪x¯aPi∪xax¯bPi∪...∪xaxb...x¯cPi,andalltheproductsaremutuallydisjoint.Severalrapidalgorithmsforcalculatingthedisjointsumhavebeenpro-posed[Locks(1987)].WeexpandtheconceptoftheSDPalgorithmtoDFTs.InaDFTcase,eachproducttermofminimalcanonicalformcon-tainsbasiceventsandorderedpairsofbasiceventslinkedbyBEFOREoperator.Theorderedpairofbasicevents(forexample,ei▹ej)canbetreatedasifitisasingleeventmeaningeventeioccursbeforebasiceventej.Astheresult,wecanapplytheSDPalgorithmtotheDFTexpression.Again,let’sconsidertheDFTinFig.3.Atfirst,inordertomakeeventoccurrencesclear,Eq.(21)isrewritten,TE=e3(e2▹e3)(e2▹e4)(e1▹e2)+(e3▹e4)(e1▹e2)(e1▹e3)+e4(e2▹e3)(e2▹e4)(e1▹e2)=e1e2e3(e2▹e3)(e2▹e4)(e1▹e2)+e1e3(e3▹e4)(e1▹e2)(e1▹e3)+e1e2e4(e2▹e3)(e2▹e4)(e1▹e2)Then,thecanonicalformistransformedtothefollowingequationbytheAbraham’stheorem.TE=e1e2e3(e2▹e3)(e2▹e4)(e1▹e2)+e1e¯2e3(e3▹e4)(e1▹e3)+e1e2e3(e3▹e4)(e1▹e2)(e1▹e3)(e2▹e3)+e1e2e3(e3▹e4)(e1▹e2)(e1▹e3)(e2▹e3)(e2▹e4)+e1e2e¯3e4(e2▹e4)(e1▹e2)Thesetermsarealreadydisjointeachother.Next,thenegativeorderedeventcanbedecomposedbyusingthefollowingrelation,(ei▹ej)=(ej▹ei)+¯eiej+¯eie¯j.Atlast,TE=e1e2e3(e2▹e3)(e2▹e4)(e1▹e2)+e1e¯2e3(e3▹e4)(e1▹e3)+e1e2e3(e3▹e4)(e1▹e2)(e1▹e3)(e2▹e3)+e1e2e¯3e4(e2▹e4)(e1▹e2).ThisisadisjointexpressionoftheTE.Thefirstonecontainsthreecutsequences,¯e4[e1,e2,e3],[e1,e2,e3,e4]and[e1,e2,e4,e3].Thesecondhase¯2[e1,e3,e4]and¯e2e¯4[e1,e3].Thethirdhasthreeterms,[e1,e3,e4,e2], 302September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book288ReliabilityModelingwithApplications[e1,e3,e2,e4]and¯e4[e1,e3,e2].Thelastshows¯e4[e1,e3,e2].TheprobabilityofeachcutsequenceisgivenbythemethodinSec.4.3.Therefore,thetopeventprobabilityisgivenbyaddingtheprobabilitiesofthese9cutsequences.5ConclusionTheDFTanalysishasbeencontinuouslystudiedinthelasttwodecades.Wereviewedtherecentextesionsoftheanalysisinthischapter.AlgebraicapproachofDFTanalysisisamethodofpromisetoextendthecapabilityofmodelingandanalyzingintermsofbothqualitativeandquantitativestandpoints.TheTEofaDFTwasrepresentedalgebraicallyasasumofproductcanonicalform.Theprobabilityofacutsequencecontainingspareeventswasformulatedundertheassumptionthatalltheeventoccurrencesinthesequencefollowexponentialfailuredistributions.TheprobabilityofTEwasderivedbyanSDPbasedmethods.Withouttheassumption,wehavetomanageaseriesofsymbolicintegrationstoobtainthesequenceprobabilityandtoexecuteourSDPprocedure.Obtainingthesequenceprobabilitywithouttheassumptionisourfuturework.ReferencesAbraham,J.A.(1979).Animprovedalgorithmfornetworkreliability,IEEETrans.onReliab.28,1,pp.58–61.Bobbio,A.,Portinale,L.,Minichino,M.andCiancamerla,E.(2001).InprovingtheanalysisofdependablesystemsbymappingfaulttreesintoBayesiannetworks,Reliab.Eng.Sys.Safety71,3,pp.249–260.Bobbio,A.andCodetta-Raiteri,D.(2004).Parametricfaulttreeswithdynamicgatesandrepairboxes,inReliab.andMainta.Sympo.,pp.459–465.Boudali,H.andDugan,J.B.(2005).AnewBayesiannetworkapproachtosolvedynamicfaulttrees,inReliab.andMainta.Sympo.,pp.451–456.Boudali,H.andDugan,J.B.(2005).Adiscrete-timeBayesiannetworkrelia-bilitymodelingandanalysisframework,Reliab.Eng.Sys.Safety,87,3,pp.337–349.Boudali,H.andDugan,J.B.(2006).Acontinuous-timeBayesiannetworkreli-abilitymodeling,andanalysisframework,IEEETrans.onReliab.,55,1,pp.86–97.Dugan,J.,Bavuso,S.andBoyd,M.(1990).FaultTreesandSequenceDependen-cies,inReliab.andMainta.Sympo.(RAMS1990),pp.286–293.Dugan,J.B.,Bavuso,S.J.andBoyd,M.A.(1992).Dynamicfault-treemodels 303September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookDynamicFaultTreeAnalysis289forfault-tolerantcomputersystems,IEEETrans.onReliab.,41,3,pp.363–377.Dugan,J.B.,Bavuso,S.J.andBoyd,M.A.(1993).FaulttreesandMarkovmodelsforreliabilityanalysisoffault-tolerantdigitalsystems,Reliab.Eng.Sys.Safety,39,3,pp.291–307.Dugan,J.B.(2000).Galileo:atoolfordynamicfaulttreeanalysis,(SpringerBerlin/Heidelberg).Dutuit,Y.andRauzy,A.(1996).Alineartimealgorithmtofindmodulesoffaulttrees,IEEETrans.onReliab.,45,3,pp.422–425.Fussell,J.B.,Henry,E.B.andMarshall,N.H.(1974).MOCUS–acomputerprogramtoobtainminimalcutsetsfromfaulttrees,ANCR-1156.Fussell,J.B.,Aber,E,FandRahl,R.G.(1976).Onthequantitativeanalysisofpriority-ANDfailurelogic,IEEETrans.onReliab.25,5,pp.324–326.Gulati,R.andDugan,J.B.(1997).Amodularapproachforanalyzingstaticanddynamicfaulttrees,inReliab.andMainta.Sympo.,pp.57–63.Lee,W.S.,Grosh,D.L.,Tillman,F.A.andLie,C.H.(1985).Faulttreeanalysis,methods,andapplications—Areview,IEEETrans.onReliab.34,3,pp.194–203.Locks,M.O.(1987).Aminimizingalgorithmforsumofdisjointproducts,IEEETrans.onReliab.,36,4,pp.445–453.Long,W.,Zhang,T.L.,Lu,Y.F.andOshima,M.(2002).OnthequantitativeanalysisofsequentialfailurelogicusingMonteCarlomethodfordifferentdistributions,ProbabilisticSafetyAssessmentandManagement(PSAM6),Elsevier,NewYork,pp.391–396.Merle,G.,Roussel,J.M.,Lesage,J.J.andBobbio,A.(2010).Probabilisticalge-braicanalysisoffaulttreeswithprioritydynamicgatesandrepeatedevents,IEEETrans.onReliab.,59,1,pp.250–261.Merle,G.,Roussel,J.M.andLesage,J.J.(2011).Dynamicfaulttreeanalysisbasedonthestructurefunction,inReliab.andMainta.Sympo.,pp.1–6.Merle,G.,Roussel,J.M.andLesage,J.J.(2011).Algebraicdeterminationofthestructurefunctionofdynamicfaulttrees,Reliab.Eng.Sys.Safety,96,2,pp.267–277.Montani,S.,Portinale,L.andBobbio,A.(2005).DynamicBayesiannetworksformodelingadvancedfaulttreefeaturesindependabilityanalysis,inESREL2005,pp.1414–1422.Montani,S.,Portinale,L.,Bobbio,A.,Varesio,M.andCodetta-Raiteri,D.(2006).Atoolforautomaticallytranslatingdynamicfaulttreesintody-namicBayesiannetworks,inReliab.andMainta.Sympo.,pp.434–441.Montani,S.,Portinale,L.,Bobbio,A.andCodetta-Raiteri,D.(2008).RADY-BAN:atoolforreliabilityanalysisofdynamicfaulttreesthroughcon-versionintodynamicBayesiannetworks,Reliab.Eng.Sys.Safety,93,7,pp.922–932.Ou,Y.andDugan,J.B.(2004).Modularsolutionofdynamicmulti-phasesys-tems,IEEETrans.onReliab.,53,4,pp.499–508.Pearl,J.(2000).Causality:Models,Reasoning,andInference(CambridgeUni-versityPress). 304September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book290ReliabilityModelingwithApplicationsRen,Y.andDugan,J.B.(1998).Designofreliablesystemsusingstaticanddynamicfaulttrees,IEEETrans.onReliab.,47,3,pp.234–244.Stamatelatos,M.(2002).Faulttreehandbookwithaerospaceapplications,NASA.Tang,Z.andDugan,J.B.(1994).Minimalcutset/sequencegenerationfordy-namicfaulttrees,inReliab.andMainta.Sympo.,pp.207–213.Vesely,W.E.(1981).Faulttreehandbook,NRC,NUREG-0492.Weber,P.andJouffe,L.(2003).ReliabilitymodellingwithdynamicBayesiannetworks,in5thIFACSympo.onFaultDetection,SupervisionandSafetyofTechnicalProcesses(SAFEPROCESS’03).Yoneda,T.,Yuge,T.,Tamura,N.andYanagi,S.(2010).Minimalcutsequencesandtopeventprobabilityofdynamicfaulttrees,in:Proc.4thAsia-PacificIntSympoonAdvancedReliab.andMainte.Modeling(APARM2010),Wellington,pp.788–795.Yuge,T.andYanagi,S.(2007).Minimalcutset/sequencesofafaulttreewithpriorityandgates,inProc.ofthe13thInt.Conf.onReliab.andQualityinDesign(ISSAT),pp.352–356.Yuge,T.andYanagi,S.(2008).Quantitativeanalysisofafaulttreewithpriorityandgates,Reliab.Eng.Sys.Safety,93,11,pp.1577–1583.Yuge,T.andYanagi,S.(2012).Bayesiannetworkmodelingfordynamicfaulttree,inProc.18thInt.Conf.onReliab.andQualityinDesign(ISSAT),Boston,pp.111–115.Yuge,T.Tamura,N.andYanagi,S.(2012).Repairablefaulttreeanalysisusingrenewalintensities,QualityTechnology&QuantitativeManagement,9,3,pp.231–241.Zhang,X.,Miao,Q.,Fan,X.andWang,D.(2009).DynamicfaulttreeanalysisbasedonPetrinets,inReliab.andMainta.Sympo.,pp.138–142. 305September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookChapter16ReliabilityAnalysisandModelingTechniqueforanOpenSourceSolutionYoshinobuTamura1andShigeruYamada21GraduateSchoolofScienceandEngineering,YamaguchiUniversity,Tokiwadai2-16-1,Ube-shi755-8611,Japan2DepartmentofSocialManagementEngineering,TottoriUniversity,Minami4-101,Koyama,Tottori680-8552,Japan1IntroductionAtpresent,thereisgrowinginterestinthenext-generationsoftwarede-velopmentparadigmbyusingnetworkcomputingtechnologiessuchasacloudcomputing.Consideringthesoftwaredevelopmentenvironment,onehasbeenchangingintonewdevelopmentparadigmssuchasconcurrentdis-tributeddevelopmentenvironmentandtheso-calledopensourceprojectbyusingnetworkcomputingtechnologies[Umar(1993)].ThesuccessfulexperienceofadoptingthedistributeddevelopmentmodelinsuchopensourceprojectsincludesGNU/Linuxoperatingsys-tem,ApacheHTTPserver,andsoon[E-SoftInc.(2012)].However,thepoorhandlingofthequalityandcustomersupportprohibitstheprogressofOSS.Wefocusontheproblemsofsoftwarequality,whichprohibittheprogressofOSS.Especially,alarge-scaleopensourcesolutionisnowat-tractingattentionasthenext-generationsoftwaredevelopmentparadigm.Also,thelarge-scaleopensourcesolutioniscomposedofseveralOSS’s.Manysoftwarereliabilitygrowthmodels(SRGM’s)[Yamada(1994)]havebeenappliedtoassessthereliabilityforqualitymanagementandtesting-progresscontrolofsoftwaredevelopment.Ontheotherhand,theeffectivemethodofdynamictestingmanagementfornewdistributed291 306September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book292ReliabilityModelingwithApplicationsdevelopmentparadigmastypifiedbytheopensourceprojecthasonlyafewpresented[MacCormacketal.(2006);Kuk(2006);ZhoumandDavis(2005);Shawetal.(2004)].Incaseofconsideringtheeffectofthedebug-gingprocessonentiresysteminthedevelopmentofamethodofreliabilityassessmentforopensourcesolution,itisnecessarytograspthesituationofregistrationforbugtrackingsystem,thecombinationstatusofOSS’s,thedegreeofmaturationofOSS,andsoon.Inthischapter,wefocusonanopensourcesolutiondevelopedunderseveralOSS’s.Wediscussausefulmethodofsoftwarereliabilityassessmentinopensourcesolutionasatypicalcaseofnext-generationdistributedde-velopmentparadigm.Then,weintroduceamethodofsoftwarereliabilityassessmentbasedonajumpdiffusionmodelbyusingstochasticdifferentialequationsinordertoconsidertheactivestateoftheopensourceprojectandthecomponentcollisionofOSS.Then,weassumethatthesoftwarefailureintensitydependsonthetime,andthesoftwarefault-reportphenomenaonthebugtrackingsystemkeepanirregularstate.Also,weanalyzeactualsoftwarefault-countdatatoshownumericalexamplesofsoftwarereliabil-ityassessmentfortheopensourcesolution.Especially,wederiveseveralreliabilityassessmentmeasuresfromourmodel.Then,weshowthatthein-troducedmodelcanassistimprovementofqualityforopensourcesolutiondevelopedunderseveralOSS.2ModelingTechnique2.1StochasticDifferentialEquationLetS(t)bethenumberofdetectedfaultsintheopensourcesolutionbytestingtimet(t≥0).SupposethatS(t)takesoncontinuousrealvalues.Sincelatentfaultsintheopensourcesolutionaredetectedandeliminatedduringthetesting-phase,S(t)graduallyincreasesasthetestingproceduresgoon.Thus,undercommonassumptionsforsoftwarereliabilitygrowthmodeling,weconsiderthefollowinglineardifferentialequation:dS(t)=λ(t)S(t),(1)dtwhereλ(t)istheintensityofinherentsoftwarefailuresattestingtimetandisanon-negativefunction.Generally,itisdifficultforuserstouseallfunctionsinopensourcesolution,becausetheconnectionstateamongopensourcecomponentsisunstableinthetesting-phaseofopensourcesolution.Consideringthe 307September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisandModelingTechniqueforanOpenSourceSolution293ComponentCollisionNoiseDETECTEDFAULTSJump【seeJumpDiffusionProcess】THECUMULATIVENUMBEROF0TIMEConfusionPeriod【seeStochasticDifferentialEquation】Fig.1Faultreportingphenomenainopensourcesolution.characteristicofopensourcesolution,thesoftwarefault-reportphenom-enakeepanirregularstateintheearlystageoftesting-phase.Moreover,theadditionanddeletionofsoftwarecomponentsarerepeatedunderthedevelopmentofanOSSsystem,i.e.,weconsiderthatthesoftwarefailureintensitydependsonthetime.Therefore,wesupposethatλ(t)inEq.(1)hastheirregularfluctuation.Thatis,weextendEq.(1)tothefollowingstochasticdifferentialequation[Arnold(1974);Wong(1971)]:dS(t)=(λ(t)+σµ(t)γ(t))S(t),(2)dtwhereσisapositiveconstantrepresentingamagnitudeoftheirregularfluctuation,γ(t)astandardizedGaussianwhitenoise,andµ(t)thecollisionlevelofopensourcecomponentasshowninFig.1.WeextendEq.(2)tothefollowingstochasticdifferentialequationofanItˆotype:{}122dS(t)=λ(t)+σµ(t)S(t)dt+σµ(t)S(t)dW(t),(3)2whereW(t)isaone-dimensionalWienerprocesswhichisformallydefinedasanintegrationofthewhitenoiseγ(t)withrespecttotimet.TheWienerprocessisaGaussianprocessandithasthefollowingproperties:Pr[W(0)=0]=1,(4)E[W(t)]=0,(5)E[W(t)W(t′)]=Min[t,t′].(6) 308September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book294ReliabilityModelingwithApplicationsByusingItˆo’sformula[Arnold(1974);Wong(1971)],wecanobtainthesolutionofEq.(3)undertheinitialconditionS(0)=vasfollows[Yamadaetal.(1994)]:[∫]tS(t)=v·expλ(s)ds+σµ(t)W(t),(7)0wherevisthetotalnumberofdetectedfaultsfortheappliedOSS’s.UsingsolutionprocessS(t)inEq.(7),wecanderiveseveralsoftwarereliabilitymeasures.Moreover,wedefinetheintensityofinherentsoftwarefailuresincaseofλ(t)=λ1(t)andλ(t)=λ2(t),andthecollisionlevelfunctionµ(t)asfollows:∫tλ1(s)ds=(1−exp[−αt]),(8)0∫tλ2(s)ds=(1−(1+αt)exp[−αt]),(9)0µ(t)=exp[−βt],(10)whereαisanaccelerationparameteroftheintensityofinherentsoftwarefailures,andβthegrowthparameterofthestabilityofopensourcesolution.Eqs.(8)and(9)meantheexponentialandS-shapedcurves.TheproposedmodelcanwidelydescribethegrowthcurvesbyusingEqs.(8)and(9).Also,wedefinethatµ(t)decreasesasthetestingproceduresgoon,i.e.,theopensourcesolutionbecomesstableasthetestingproceduresgoon.2.2JumpDiffusionProcessThejumptermcanbeaddedtotheintroducedstochasticdifferentialequa-tionmodelsinordertoincorporatetheirregularstatearoundthetimetbyachangeinthelevelofcomponentcollision.Then,thejump-diffusionprocess[Merton(1976)]isgivenasfollows.{}122dSj(t)=λ(t)+σµ(t)S(t)dt+σµ(t)S(t)dW(t)2M∑t(λ)+d(Vi−1),(11)i=1whereMt(λ)isaPoissonpointprocesswithparameterλatoperationtimet.Also,Mt(λ)thenumberofoccurredjumps,λthejumprate.Mt(λ), 309September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisandModelingTechniqueforanOpenSourceSolution295W(t),andViareassumedtobemutuallyindependent.Moreover,Viisi-thjumprange.dSj(t)meansthatthejumptermisaddedtodS(t),i.e.,dSj(t)hasthemixeddistributioncomposedofboththeWienerprocessandthejumpdiffusionprocess.ByusingItˆo’sformula[Arnold(1974);Wong(1971)],thesolutionoftheformerequationcanbeobtainedasfollows:∫tM∑t(λ)Sj(t)=v·expλ(s)ds+σµ(t)W(t)−logVi.(12)0i=13ParameterEstimation3.1MethodofMaximum-likelihoodInthissection,theestimationmethodofunknownparametersα,βandσinEq.(7)ispresented.LetusdenotethejointprobabilitydistributionfunctionoftheprocessS(t)asP(t1,y1;t2,y2;···;tK,yK)≡Pr[N(t1)≤y1,···,N(tK)≤yK|S(t0)=v],(13)whereS(t)isthecumulativenumberoffaultsdetecteduptothetestingtimet(t≥0),anddenoteitsdensityasp(t1,y1;t2,y2;···;tK,yK)∂KP(t,y;t,y;···;t,y)1122KK≡.(14)∂y1∂y2···∂yKSinceS(t)takesoncontinuousvalues,weconstructthelikelihoodfunctionlfortheobserveddata(tk,yk)(k=1,2,···,K)asfollows:l=p(t1,y1;t2,y2;···;tK,yK).(15)Forconvenienceinmathematicalmanipulations,weusethefollowingloga-rithmiclikelihoodfunction:L=logl.(16)Themaximum-likelihoodestimatesα∗,β∗andσ∗arethevaluesmakingLinEq.(16)maximize.Thesecanbeobtainedasthesolutionsofthe 310September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book296ReliabilityModelingwithApplicationsfollowingsimultaneouslikelihoodequations[Yamadaetal.(1994)]:∂L∂L∂L===0.(17)∂α∂β∂σ3.2EstimationofJump-diffusionParametersGenerally,itisdifficulttoestimatethejump-diffusionparametersofstochasticdifferentialequationmodelbecauseofthecomplicatedlikeli-hoodfunction,mixeddistribution,etc.Theestimationmethodsofjump-diffusionparametersareintroducedbyseveralresearchers.However,theeffectivemethodofestimationhasonlyafewpresented.Wefocusontheestimationmethodsperformedintwostages[Honor´e(1998)].Ageneticalgorithm(GA)inordertoestimatethejump-diffusionparametersoftheintroducedmodelisusedinthissection.TheprocedureofGAalgorithmisgiveninthefollowing[Holland(1975)].Itisassumedthattheintroducedjump-diffusionmodelincludestheparametersλ,µ,andτ.Theparametersµandτmeantheparametersincludedini-thjumprangeVi.Step1Theinitialindividualsarerandomlygenerated.Also,thesetofinitialindividualisconvertedtothebinarydigit.Step2Twoparentalindividualsareselected,andnewindividualsareproducedbythecrossoverrecombination.Step3Thevalueoffitnessiscalculatedfromtheevaluatedvalueofeachindividual.Thefollowingvalueoffitnessastheerrorbetweentheestimatedandtheactualvaluesisdefinedinthispaper:minFi(),∑K2Fi={Sj(i)−yi},(18)i=0whereSj(i)isthenumberofdetectedfaultsatoperationtimeiintheintroducedjump-diffusionmodel,yithenumberofactualdetectedfaults.Also,meansthesetofparametersλ,µ,andτ.Step4Step2andStep3arecontinueduntilreachingthespecificsize.Thejump-diffusionparametersλ,µ,andτareestimatedbyusingabovementionedsteps. 311September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisandModelingTechniqueforanOpenSourceSolution2974SoftwareReliabilityAssessmentMeasures4.1StochasticDifferentialEquationModelWeconsidertheexpectednumberoffaultsdetecteduptotestingtimet.ThedensityfunctionofW(t)isgivenby:[]1W(t)2f(W(t))=√exp−.(19)2πt2tInformationonthecumulativenumberofdetectedfaultsinthesystemisimportanttoestimatethesituationoftheprogressonthesoftwaretestingprocedures.SinceS(t)isarandomvariableinourmodel,itsexpectedvaluecanbeausefulmeasure.WecancalculatethemfromEq.(7)asfollows[Yamadaetal.(1994)]:[∫]tσ2µ(t)2E[S(t)]=v·expλ(s)ds+t,(20)02whereE[S(t)]istheexpectednumberoffaultsdetecteduptotimet.Also,theexpectednumberofremainingfaultsattimetcanobtainasfollows:E[N(t)]=v·e−E[S(t)],(21)wherevisthetotalnumberofdetectedfaultsfortheappliedOSS’s,andemeansexp(1).4.2Jump-diffusionModelSimilarly,thecumulativenumberofdetectedfaultsinthesystemisimpor-tanttoestimatethesituationoftheprogressonthesoftwaredebuggingprocedures.SinceNj(t)isarandomvariableintheintroducedmodel,itiscalculatedasEq.(12)[Yamadaetal.(1994)].Also,itisimportantforsoftwaremanagerstoassessthenumberoflatentfaultsaccordingtothechangeofspecificationofOSS.Thenumberofremainingfaultsbasedonthejump-diffusionmodelconsideringthechangeofrequirementsspecificationcanbeobtainedasfollows:Nj(t)≃v·e−Sj(t).(22)Also,themeantimebetweensoftwarefailuresisusefultomeasurethepropertyofthefrequencyofsoftwarefailure-occurrences.Then,thecumu-lativeMTBF(denotedbyMTBFC)isapproximatelygivenby:tMTBFCj(t)≃.(23)Sj(t) 312September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book298ReliabilityModelingwithApplications5NumericalIllustrations5.1DataforNumericalIllustrationsWefocusonalarge-scaleopensourcesolutionbasedontheApacheHTTPServer[TheApacheHTTPServerProject(2012)],ApacheTomcat[ApacheTomcat(2012)],MySQL[MySQL(2012)]andJSP(JavaServerPages).Thefault-countdatausedinthispaperarecollectedinthebugtrackingsystemonthewebsiteofeachopensourceproject.5.2ReliabilityAssessmentResultsTheestimatedexpectedcumulativenumbersofdetectedfaultsinEq.(20),E[bS(t)]’s,incaseofλ(t)=λ1(t)andλ(t)=λ2(t)areshowninFigs.2and3,respectively.Also,thesamplepathsoftheestimatednumbersofdetectedfaultsinEq.(7),Sb(t)’s,incaseofλ(t)=λ1(t)andλ(t)=λ2(t)areshowninFigs.4and5,approximately.Moreover,thesamplepathsoftheesti-matednumbersofdetectedfaultsinEq.(7),Scj(t)’s,incaseofλ(t)=λ1(t)andλ(t)=λ2(t)areshowninFigs.4and5,approximately.Furthermore,theestimatedexpectedcumulativenumbersofremainingfaultsinEq.(21),E[bN(t)]’s,incaseofλ(t)=λ1(t)andλ(t)=λ2(t)areshowninFigs.8and9,respectively.1008060DATAActualEstimate40DETECTEDFAULTS20CUMULATIVENUMBEROF00510152025TIME(WEEKS)Fig.2Estimatedcumulativenumberofdetectedfaults,E[ˆS(t)],incaseofλ(t)=λ1(t). 313September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisandModelingTechniqueforanOpenSourceSolution2991008060DATAActualEstimate40DETECTEDFAULTS20CUMULATIVENUMBEROF00510152025TIME(WEEKS)Fig.3Estimatedcumulativenumberofdetectedfaults,E[ˆS(t)],incaseofλ(t)=λ2(t).1008060DATAActualSamplePath40DETECTEDFAULTS20CUMULATIVENUMBEROF00510152025TIME(WEEKS)Fig.4Samplepathoftheestimatednumberofdetectedfaults,Sb(t),incaseofλ(t)=λ1(t). 314September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book300ReliabilityModelingwithApplications1008060DATAActualSamplePath40DETECTEDFAULTS20CUMULATIVENUMBEROF00510152025TIME(WEEKS)Fig.5Samplepathoftheestimatednumberofdetectedfaults,Sb(t),incaseofλ(t)=λ2(t).1008060DATASamplePathActual40DETECTEDFAULTS20CUMULATIVENUMBEROF00510152025TIME(WEEKS)Fig.6Samplepathoftheestimatednumberofdetectedfaults,Scj(t),incaseofλ(t)=λ1(t). 315September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisandModelingTechniqueforanOpenSourceSolution3011008060DATASamplePathActual40DETECTEDFAULTS20CUMULATIVENUMBEROF00510152025TIME(WEEKS)Fig.7Samplepathoftheestimatednumberofdetectedfaults,Scj(t),incaseofλ(t)=λ2(t).1008060DATAActualEstimate40REMAININGFAULTS20CUMULATIVENUMBEROF0051015202530TIME(WEEKS)Fig.8Estimatedcumulativenumberofremainingfaults,E[ˆS(t)],incaseofλ(t)=λ1(t). 316September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book302ReliabilityModelingwithApplications1008060DATAActualEstimate40REMAININGFAULTS20CUMULATIVENUMBEROF0051015202530TIME(WEEKS)Fig.9Estimatedcumulativenumberofremainingfaults,E[ˆS(t)],incaseofλ(t)=λ2(t).0.50.40.3DATAEstimate0.2CumulativeMTBF0.10.0051015202530TIME(WEEKS)Fig.10EstimatedMTBFC,MTBF\C(t),incaseofλ(t)=λ1(t). 317September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisandModelingTechniqueforanOpenSourceSolution3030.50.40.3DATAEstimate0.2CumulativeMTBF0.10.0051015202530TIME(WEEKS)Fig.11EstimatedMTBFC,MTBF\C(t),incaseofλ(t)=λ2(t).1.00.80.6DATAEstimate0.40.2COEFFICIENTOFVARIATION0.0051015202530TIME(WEEKS)Fig.12Estimatedcoefficientofvariation,CV(t),incaseofλ(t)=λ1(t). 318September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book304ReliabilityModelingwithApplications1.00.80.6DATAEstimate0.40.2COEFFICIENTOFVARIATION0.0051015202530TIME(WEEKS)Fig.13Estimatedcoefficientofvariation,CV(t),incaseofλ(t)=λ2(t).TheestimatedMTBFC’sincaseofλ(t)=λ1(t)andλ(t)=λ2(t)areplottedinFigs.10and11,respectively.ThesefiguresshowthattheMTBFCincreaseasthetestingproceduresgoon.Also,CV(t)’sincaseofλ(t)=λ1(t)andλ(t)=λ2(t)areshowninFigs.12and13,respectively.FromFigs.12and13,wecanconfirmthattheestimatedcoefficientofvariationapproachestheconstantvalue.Weconsiderthatthecoefficientofvariationisusefulmeasuretocompareseveralfaultdatasetsinthepastsystemdevelopmentprojects.Fromabovementionedresults,wecanconfirmthattheourmodelcancoverthenoiseofcollisionlevelofopensourcesolutionbyusingtheWienerprocessandthejumpdiffusionprocess.6ConcludingRemarksInthischapter,wehavefocusedontheopensourcesolutionwhichisknownasthelarge-scalesoftwaresystem,anddiscussedthemethodofreliabilityassessmentfortheopensourcesolutiondevelopedunderseveralOSS’s.Moreover,wehaveintroducedasoftwarereliabilitygrowthmodelbasedonajumpdiffusionprocessbyusingstochasticdifferentialequationsin 319September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisandModelingTechniqueforanOpenSourceSolution305ordertoconsidertheactivestateoftheopensourceprojectandthecollisionamongtheopensourcecomponents.Especially,wehaveassumedthatthesoftwarefailureintensitydependsonthetime,andthesoftwarefault-reportphenomenaonthebugtrackingsystemkeepanirregularstate.Also,wehaveanalyzedactualsoftwarefault-countdatatoshownumericalexamplesofsoftwarereliabilityassessmentforthelarge-scaleopensourcesolution.Moreover,wehavederivedseveralreliabilityassessmentmeasuresfromourmodel.Atpresent,anewparadigmofdistributeddevelopmenttypifiedbysuchopensourceprojectwillevolveatarapidpaceinthefuture.Especially,itisdifficultforthesoftwaretestingmanagerstoassessthereliabilityforthelarge-scaleopensourcesolutionasatypicalcaseofnext-generationdis-tributeddevelopmentparadigm.Ourmethodmaybeusefulasthemethodofreliabilityassessmentforthelarge-scaleopensourcesolution.AcknowledgmentsThisworkwassupportedinpartbytheGrant-in-AidforScientificRe-search(C),GrantNo.24500066andNo.25350445fromtheMinistryofEducation,Culture,Sports,Science,andTechnologyofJapan.ReferencesApacheTomcat,TheApacheSoftwareFoundation.[Online].Available:http://tomcat.apache.org/Arnold,L.(1974).StochasticDifferentialEquations-TheoryandApplications,JohnWiley&Sons,NewYork.E-SoftInc.,InternetResearchReports.[Online].Available:http://www.securityspace.com/ssurvey/data/Holland,J.H.(1975).AdaptationinNaturalandArtificialSystems,UniversityofMichiganPress.Honor´e,P.(1998).Pitfallsinestimatingjump-diffusionmodels,WorkingPaperSeries18,UniversityofAarhus,SchoolofBusiness.Kuk,G.(2006).StrategicinteractionandknowledgesharingintheKDEdevel-opermailinglist,InformsJournalofManagementScience,vol.52,no.7,pp.1031–1042.LiP.,Shaw,M.,Herbsleb,J.,Ray,B.,andSanthanamP.(2004).Empiricaleval-uationofdefectprojectionmodelsforwidely-deployedproductionsoftwaresystems,Proceedingofthe12thInternationalSymposiumontheFounda-tionsofSoftwareEngineering(FSE-12),pp.263–272. 320September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book306ReliabilityModelingwithApplicationsMacCormack,A.,Rusnak,J.,andBaldwin,C.Y.(2006).Exploringthestruc-tureofcomplexsoftwaredesigns:anempiricalstudyofopensourceandproprietarycode,InformsJournalofManagementScience,vol.52,no.7,pp.1015–1030.Merton,R.C.(1976).Optionpricingwhenunderlyingstockreturnsarediscon-tinous,JournalofFinancialEconomics,vol.3,pp.125–144.MySQL,OracleCorporationand/oritsaffiliates.[Online].Available:http://www.mysql.com/TheApacheHTTPServerProject,TheApacheSoftwareFoundation.[Online].Available:http://httpd.apache.org/Umar,A.(1993).DistributedComputingandClient-ServerSystems,PrenticeHall,EnglewoodCliffs,NewJersey.Wong,E.(1971).StochasticProcessesinInformationandSystems,McGraw-Hill,NewYork,1971.Yamada,S.(1994).SoftwareReliabilityModels:FundamentalsandApplications(inJapanese),JUSEPress,Tokyo.Yamada,S.,Kimura,M.,Tanaka,H.,andOsaki,S.(1994).Softwarereliabilitymeasurementandassessmentwithstochasticdifferentialequations,IEICETransactionsonFundamentals,vol.E77–A,no.1,pp.109–116.Zhoum,Y.andDavis,J.(2005).Opensourcesoftwarereliabilitymodel:anempiricalapproach,ProceedingsoftheWorkshoponOpenSourceSoftwareEngineering(WOSSE),vol.30,no.4,pp.67–72. 321September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookChapter17MaintenanceModelsofMiscellaneousSystemsKodoItoInstituteofConsumerSciencesandHumanLife,KinjoGakuinUniversity,1723Omori2-chome,Moriyama-ku,Nagoya463–8521,Japan1IntroductionMiscellaneoussystemssuchassocialinfrastructureandmasstransporta-tion,sustainourcomfortabledailylivesandeconomicactivities.Forthestableoperationofthesesystemswithoutanyseverefaults,suitablemain-tenanceshavetoundergo.However,maintenancecostsbecomeextraor-dinarilypricyinmostadvancedcountriesbecauseofexpensivepersonnelexpenditure.Today,theconflictbetweentheausteritybudgetandtheinevitablesustenancedemandbecomesaserioussocialissueandtheestab-lishingcost-effectivemaintenancehasbecomeatopprioritysocialmatter.Themaintenanceisclassifiedintopreventivemaintenance(PM)andcorrectivemaintenance(CM):PMisamaintenancepolicyinwhichweundergoessomemaintenanceonaspecificschedulebeforefailure,andCMisamaintenancepolicyafterfailure[BarlowandProschan(1965);Nakagawa(2005)].ManyresearchershavestudiedoptimalPMpoliciesbecausetheCMcostisusuallymuchhigherthanthePMoneandtheoptimalPMpolicydiffersineachindividualsystem.Therefore,targetsystemcharacteristicsmustbeinvestigatedminutelyfortheconsiderationofcost-effectivePMpolicies.Inthischapter,wesurveyoptimalmaintenancemodelsfortwodifferentsystemssuchastheagedfossil-firedpowerplantandthecivilaircraftbasedonouroriginalworks.307 322September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book308ReliabilityModelingwithApplicationsInSection2,weconsidertheagedfossil-firedpowerplantmainte-nance:Agedfossil-firedpowersystems,whichneedthemaintenancefortheirsteadyoperations,areonthegreatincreaseinJapan.Thepreventivemaintenanceand/orrepairofsuchsystemsareindispensabletopreventtheserioustroublesuchastheemergencystopofoperation.Becausethecumulativedamageofsystempartsremains,theconditionofsystemafterrepaircannotreturntobrand-new.Suchrepairdegradationofsystemhavetobeconsideredwhenthemaintenanceplanisestablished.InSection2,asystemisrepairedatprespecifiedschedulewhenthecumulativedamagelevelisbelowamanageriallevel.Whenthecumulativedamagelevelex-ceedsacertaincriticallevel,thesystemfailsandsuchcriticallevellowersateveryrepair.Theexpectedcostperunitoftimebetweenmaintenancesissecured,andtheoptimalmaintenancepolicyisderived.Section3isdevotedtothecomparisonofthreecumulativedamagemodels:(1)Aunitissubjectedshocksandsufferssomedamageduetoshocks(Model1).(2)Theamountofdamageduetoshocksismeasuredonlyatperiodictimes(Model2).(3)Theamountofdamageincreaseslinearlywithtime(Model3).Thetotaldamageisadditiveandtheunitfailswhenthetotaldamagehasexceededafailurelevelforthreemodels.Models2and3wouldbeactuallyusedastheapproximatedonesofModel1.Asthepreventivereplacementpolicy,theunitisreplacedbeforefailureataplannedtime.Theexpectedcostratesofeachmodelareobtained,andoptimalpoliciesthatminimizethemarederived.Finally,Section4takesupthecivilaircraftmaintenance:Astheac-cumulationoftinystresscausesthefailureofairframe,itsmaintenanceisindispensabletooperateaircraftwithoutanyserioustroubles.Fourech-elonspreventivemaintenance(PM)isundergoneforcommercialaviationandistheimperfectmaintenance.TheimperfectmaintenanceassumesthereductionrateofcumulativehazardfunctionsafterPMsandhowtosettlethisrateisaproblem.InSection4,astandardimperfectpreventivemain-tenancepolicy(Model1)isdiscussed.AnditismodifiedtoModel2whichisusefulinactualdesign.Theairframefailureduringoperationisnotsoseriousandthefailurerateremainsundisturbedbyrepair.TheexpectedcostrateofairframemaintenanceissecuredandtheoptimalPMnumberwhichminimizesitisdiscussed.Weobtaintheexpectedcostsortheavailabilityofeachmodelasanobjectivefunctionandderiveoptimalmaintenancepolicieswhichminimizethem,usingreliabilitytechniques.Furthermore,wegiveshortlysomecom-mentsregardingthelimitationandpossibleextensionsoftheabovemodels. 323September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookMaintenanceModelsofMiscellaneousSystems3092OptimalMaintenancePolicyforaDamageSystemwithRepairAnumberofagedfossil-firedpowerplantsareincreasinginJapan.Forex-ample,33%oftheseplantsarecurrentlyoperatedfrom150,000to199,999hours(from17to23years),and26%ofthemareabove200,000hours(23years)[Hisano(2000)].AlthoughJapanesegovernmenteliminatesreg-ulationsofelectricpowerindustry,mostindustriesrestrainfromthein-vestmentfornewplantsandareprefertooperatecurrentplantsefficientlybecauseofthelong-termrecessionandthereducingofoperatingatomicreactornumberinJapan.Thedeliberativemaintenanceplansareindispensabletooperatetheseagedplantswithoutserioustroublesuchastheemergencystopofopera-tion.Theimportanceofmaintenanceforagedplantsismuchhigherthanthatfornewonesbecauseoccurrenceprobabilitiesofseveretroublesin-creaseandnewfailurephenomenamightappearaccordingtothedegra-dationofplants.Furthermore,actuallifespansofplantcomponentsaremostlydifferentfrompredictedonesbecausetheyareaffectedbyvariouskindsoffactorssuchasmaterialqualitiesandoperationalcircumstances[Hisano(2001)].So,maintenanceplansshouldbeestablishedconsideringoccurrenceprobabilitiesofmiscellaneouscomponents.Theoccurrenceoffailureisdiscussedbyutilizingthecumulativedamagemodel.Ithasbewell-knownthatthefollowingstochasticreliabilitymodeliscalledshockorcumulativedamagemodel:Aunitissubjectedtoshocksandsufferssomeamountofdamagesuchaswear,fatigue,crackgrowth,creep,anddielectricateachshock.Thetotaldamageduetoshocksisaddi-tive.Thiscanbedescribedtheoreticallybycumulativeprocess[Cox(1962)]andcompoundPoissonprocesses[C¸inlar(1975);Ross(1983)].Somereli-abilityquantitieswereobtainedin[Esary,MarshallandProschan(1973);NakagawaandOsaki(1974)].Markovmodelsofsuchmodelswerestudiedandtheirlifedatawerecollectedin[BogdanoffandKozin(1985)].Sev-eralkindsofdamagemodelswereanalyzed,usingcumulativeprocesses,andtheirmaintenancepoliciesweresummarizedandoptimalpolicieswerediscussedanalyticallyin[Nakagawa(2007)].Aplantconsistsofawidevarietyofmechanicalpartssuchaspowerboiler,compressor,combustor,steamandgasturbines.Somepartssuf-fershightemperatureatoperationandsuchthermaldamagesaccumu-latedintheseparts.PMisperformedperiodicallybeforethesedamagescauseseriousfailures.TheconditionofsystemafterPMcannotreturnto 324September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book310ReliabilityModelingwithApplicationsthebrand-newconditionbecausethecumulativefatiguedamageofsystempartsremainsafterPM[Kosugiyama,Takizuka,Kunitomi,Yan,KatanishiandTakada(2003)].InpastPMstudiesandcumulativedamagemodels,theconditionofsystemafterPMissupposedtobebrand-new.Intheactualplantmain-tenance,theremainingdamageafterPMshouldbeconsidered.Wecon-sideredacumulativedamagePMmodelinwhichasystemisrepairedatprespecifiedschedulewhenthecumulativedamagelevelisbelowaman-ageriallevelandthedamageremainsaftertherepair[Ito,NakagawaandTeramoto(2006)].Weextendedittheoptimaloperationcencoringpolicywhichmamimizestheexpectedprofitofsystem[ItoandNakagawa(2006,2007)].Inthissection,thecumulativedamagePMmodelisconsidered[Ito,NakagawaandTeramoto(2006)].Whenthecumulativedamagelevelex-ceedsacertaincriticallevel,thesystemfailsandthecriticalleveldegradesateveryrepair.Theexpectedcostperunitoftimebetweenmaintenanceisconsideredandtheoptimalmaintenancepolicyisderived.2.1Model1Weconsiderthefollowingmaintenancepolicy:1)Thesystemisoperatingcontinuouslyandshocksduringoperationoc-curataPoissonprocess.Theprobabilitythatthej-thshockoccursduring(0,t]isH(t)=[(λt)j/j!]e−λt(j=0,1,2,···)[Osaki(1992)].jThus,theprobabilitythatshocksoccurmorethanj-timesduring(0,t]is∑∞Fj(t)=i=jHi(t).2)Thedamagecausedbyeachshockhasanidenticalprobabilitydistribu-tionG(x)≡Pr{Yj≤x}(j=1,2,···)withfinitemean,andeachdam-∑jageisadditive.Then,thetotaldamageZj≡i=1Yitothej-thshockwhereZ≡0hasadistributionPr{Z≤x}=G(j)(x)(j=1,2,···),0jwhereΦ(j)(x)(j=1,2,···)denotesthej-foldStieltjesconvolutionof(0)∑∞(j)Φ(x)withitselfandΦ(x)≡1forx≥0.MΦ(x)≡j=1Φ(x)isarenewalfunctionofanydistributionΦ(x).3)Thetotaldamageisbelowamanageriallevelkduring(Ti,Ti+1](i=0,1,2,···)whereT0≡0,thesystemisrepairedattimeTi+1anditscostisc0.Thesystemdegradesateveryrepair.4)WhenthetotaldamageexceedsafailurelevelKi(i=0,1,···),thesys-temfailsanditsmaintenancecostiscKi,whereKideclines,i.e.,K0> 325September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookMaintenanceModelsofMiscellaneousSystems311Fig.1SchematicdiagramofModel1.K1>···>Ki−1>Ki>···becausethedamageofsystempartsremainsateveryrepair(seeFig.1).5)WhenthetotaldamageisbetweenkandKi,thesystemisoverhauledanditscostisc1(cKi>c1>c0).Inthismodel,thesystemoperatesuntilitsdamageexceedskwhichislessthanKi,i.e.,limi→∞Ki>k.TheprobabilityPkithatthesystemundergoesoverhaulwhenthetotaldamageexceedskis∑∞∫Ti+1P=[G(j−1)(k)−G(j)(k)]dF(t),(1)kijj=1Ti∑∞anditisobviousthati=0Pki=1.TheprobabilityPKithatthesystemfailsduring(Ti,Ti+1]whenthetotaldamageexceedsKi,is∑∞∫k∫Ti+1P=[1−G(K−x)]dG(j−1)(x)dF(t).(2)Kiijj=10TiLetE{U}denotethemeantimetosomemaintenance.From(1)and(2),∑∞∑∞∫Ti+1E{U}=[G(j−1)(k)−G(j)(k)]tdF(t)ji=0j=1Ti1=[1+MG(k)].(3)λ 326September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book312ReliabilityModelingwithApplicationsFurther,thetotalexpectedcostE{C}tosomemaintenanceis∑∞E{C}=[(c1+ic0)(Pki−PKi)+(cKi+ic0)PKi].(4)i=0Therefore,from(3)and(4),theexpectedcostrateis∑∑{∫T∞∞(c+ic)[G(j−1)(k)−G(j)(k)]i+1dF(t)i=0j=110Tij}∫k(j−1)∫Ti+1C1(k)+(cKi−c1)0[1−G(Ki−x)]dG(x)TidFj(t)=.λ1+MG(k)(5)2.1.1Case1SupposethatG(x)=1−exp(−µx),i.e.,G(j)(x)=∑∞[(µx)i/i!]e−µxandi=jMG(x)=µx.Then,(4)is∑∞∑∞E{C}=c−c+[(A−A)eµk+c]G(j)(k)H(T),(6)10ii−10jii=0j=0whereA≡(c−c)e−µKi(i=0,1,2,···)andA≡0.Therefore,fromiKi1−1(3)and(6),theexpectedcostrateis∑∞µk∑∞(j)C1(k)c1−c0+i=0[(Ai−Ai−1)e+c0]j=0G(k)Hj(Ti)=.(7)λ1+µkDifferentiatingC1(k)withrespecttokandputtingittozero,∑∞∑∞{[](A−A)eµk+c(1+µk)G(j−1)(k)ii−10i=0j=0[]}−(A−A)eµk+c(2+µk)G(j)(k)H(T)=c−c,(8)ii−10ji10whereG(j)(k)≡0(j<0).Lettingdenotetheleft-handsideof(8)byL1(k),∑∞L(0)={[(A−A)+c](λT−1)−c}e−λTi,(9)1ii−10i0i=0∑∞∑∞{[]L(K)=(A−A)eµK∞+c(1+µK)G(j−1)(K)1∞ii−10∞∞i=0j=0[]}−(A−A)eµK∞+c(2+µK)G(j)(K)H(T).(10)ii−10∞∞ji 327September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookMaintenanceModelsofMiscellaneousSystems313Thus,ifL(0) 328September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book314ReliabilityModelingwithApplicationsFig.2SchematicdiagramofModel2.Next,thetotalexpectedcosttosomemaintenanceortimeTnisn∑−1E{C}=[c1+(n−1)c0]PTn+(c1+ic0)(Pki−PKi)i=0n∑−1+(cKi+ic0)PKii=0∑n∑∞=c−c+[(A−A)eµk+c]G(j)(k)H(T)10ii−10jii=0j=0∑∞−AeµkG(j)(k)H(T).(15)n−1jnj=0Therefore,from(14)and(15),theexpectedcostrateis∑nµkc1−c0+i=0[(Ai−Ai−1)e+c0]×∑∞G(j)(k)H(T)−Aeµk∑∞G(j)(k)H(T)j=0jin−1j=0jnC2(n,k)=∑∞∫Tn,G(j)(k)Hj(t)dtj=00(16)whichagreeswith(7)asn→∞. 329September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookMaintenanceModelsofMiscellaneousSystems3153ComparisonofThreeCumulativeDamageModelsInSection2,thecumulativedamagemodelisutilizedforestablishingtheoptimalPMpoliciesofagedfossil-firedpowerplant.Inthissection,weexpandthecumulativedamagemodel[ItoandNakagawa(2008)].First,weconsiderastandardcumulativedamagemodelwheretheunitsufferssomedamageduetoshocksandthetotaldamageisadditive.TheunitfailswhenthetotaldamagehasexceededafailurelevelK.Theprob-abilitythattheunitfailsintimetanditspropertiesareobtainedbycu-mulativeprocesses[Nakagawa(2007)].However,itmightbeimpossibletoestimateandknowoccurrencesofshocksandthetotaldamageeveryateachshock.Secondly,theamountofdamageduetoshocksismeasuredonlyatperiodictimesbyinspections,irrespectiveoccurencesofshocks.Thirdly,thetotaldamageincreaseslinearlywithtimet.Thedistributionsoftotaldamageanditsmeansforthreemodelsareobtained.Next,weconsideranagereplacementpolicywheretheunitisreplacedwhenthetotaldamagehasexceededafailurelevelKorattimeTforModels1&3andNT0forModel2,whicheveroccursfirst.Theexpectedcostratesofthreemodelsareobtainedandoptimalpoliciesthatminimizethemarederived.3.1ThreeModelsWeconsiderthefollowingthreecumulativedamagemodelsforanoperatingunit[Cox(1962);Nakagawa(2007)]:3.1.1Model1:StandardmodelTheunitissubjectedtoshocksandsufferssomedamageateachshock.LetrandomvariablesXj(j=1,2,···)denoteasequenceofinterarrivaltimesbetweensuccessiveshocks,andrandomvariablesWj(j=1,2,···)denotethedamageproducedbythejthshock,whereW0≡0.Further,letN(t)denotetherandomvariablethatisthetotalnumberofshocksuptotimet.Then,definearandomvariableN∑(t)Z(t)≡Wj(N(t)=0,1,2,···),(17)j=0thatrepresentsthetotaldamageattimet.ItisassumedthatF(t)≡Pr{Xj≤t}withfinitemean1/λandvarianceσ2andG(x)≡P{W≤x}withfinitemeanµandvarianceσ2fort,x≥0.FrjG 330September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book316ReliabilityModelingwithApplicationsThen,theprobabilitythatshocksoccurexactlyjtimesin[0,t]isP{N(t)=j}=F(j)(t)−F(j+1)(t)(j=0,1,2,···),(18)r∑∞P{Z(t)≤x}=G(j)(x)[F(j)(t)−F(j+1)(t)].(19)rj=0Thus,themeantotaldamageattimetis∑∞E{Z(t)}=µF(j)(t)=µM(t),(20)Fj=1anditsapproximatevarianceofZ(t)is[Nakagawa(2007)]()σ2V{Z(t)}≈λ3tµ2σ2+G.(21)Fλ23.1.2Model2:PeriodicmodelEachamountWn(n=1,2,···)ofdamageismeasuredonlyatperiodictimesnT(n=1,2,···)foragivenT0(0 331September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookMaintenanceModelsofMiscellaneousSystems3173.2OptimalReplacementPoliciesSupposethattheunitfailswhenthetotaldamagehasexcceededafailurelevelK.Asthepreventivereplacementpolicy,theunitisalsoreplacedbeforefailureattimeT.Fortheabovereplacementmodel,costsc1andc2arethereplacementcostsatfailureandattimeT,wherec1>c2.TheexpectedcostrateforModel1is[Nakagawa(2007)]c−(c−c)∑∞[F(j)(T)−F(j+1)(T)]G(j)(K)112j=0C1(T)=∑∞∫T.(28)G(j)(K)[F(j)(t)−F(j+1)(t)]dtj=00TheoptimaltimeT∗thatminimizesC(T)isasfollows:Letf(t)bea11densityfunctionofF(t)and∑∞f(j+1)(T)[G(j)(K)−G(j+1)(K)]j=0Q1(T)≡∑∞(j)(j+1)(j).[F(T)−F(T)]G(K)j=0Then,ifQ1(T)increasesstrictlywithTandQ1(∞)[1+MG(K)]>λc1/(c1−c),thenthereexistsafiniteanduniqueT∗(0 332September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book318ReliabilityModelingwithApplicationsFinally,forModel3,theunitisreplacedattimeToratfailure,whicheveroccursfirst.WhenZ(t)=Att,theexpectedcostrateisc1−(c1−c2)LA(K/T)C3(T)=∫T.(32)LA(K/t)dt0TheoptimaltimeT∗thatminimizesC(T)is:Letl(t)beaden-33AsityfunctionofLA(t)andr(t)bethefailurerateofLA(t),i.e.,r(t)≡lA(t)/LA(t).Then,ifr(t)increaseswitht,thereexistsafiniteanduniqueT∗thatsatisties3∫Tc1r(K/T)LA(K/t)dt+LA(K/T)=.(33)0c1−c24OptimalImperfectMaintenanceofAircraftTheairframeofcommercialaviationhastobelightweightbecauseitsweightaffectstherangeandfuel-efficiencyofaircraft.Coincidentally,theairframehastobedamagetolerantduringoperationbecauseairframesuffersseri-ousstatisticalanddynamicmechanicalstresswhichiscausedbythevari-ationofenvironmentalairpressureandtemperature,vibrationofengineandaerodynamicturbulenceandshockswhentheaircrafttakesoffandlands.Althoughtheferrousalloyiscommonmetallicmaterialformechan-icalstructure,aluminumalloys2024and7075areutilizedforthemajormaterialofairframestructurebecausethetensilestrengthofaluminumalloyisalmostassameasthatofferrousalloyandthespecificweightofaluminumalloyisfromonethirdtoonefourthofferrousalloy.Further-more,structuralconsiderationforlightweightisperformedwhenairframeisdesigned.Thestressedskinmonocoquestructureisadoptedanditsstressisanalyzedstrictlyusingfiniteelementmethod(FEM)forreducingweight[Paul,KellyandVenkayya(2002)].Undercyclicstresscondition,theconsumedlifeofmechanicalstructuresisassessedbyS-Ncurve,andtheferrousalloystructurehasinfinitelife-timewhenitisdesignedtoholdstressbelowthresholdofendurancelimit.Whereas,non-ferrousalloysuchasaluminumhasnosuchdistinctlimitandfailsfinallyunderslightstresscondition.Thus,thelifetimeofaluminumalloyairframeisfiniteandtheaccumulationoftinystresseswillcausese-riousdamageinalongperiod.Therefore,themaintenanceofairframeisindispensabletooperateaircraftwithoutanyserioustroubles.InSection1,wedenotedthatdetailedinvestigationsoftargetsystemcharacteristicsmustbeperformedfortheconsiderationofcost-effective 333September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookMaintenanceModelsofMiscellaneousSystems319PMpolicies.Toundergoingthecost-effectivePM,variouskindsofPMsofwhichtimeperiodsandmaintenancecontentsaredifferent,areassembledappropriately.Wecallsuchmaintenanceasthemulti-echelonPMandstudiedoptimalmaintenancepoliciesforthefossil-firedpowerplantwithmulti-echelonrisks[NakagawaandIto(2008)].FollowingfourechelonsPMsareundergoneforcommercialaviation[FAAAdvisoryCircular(1998)]:1)Acheck:Inspectionoflandinggear,controlsurfaces,fluidlevels,oxy-gensystems,lightingandauxiliarypowersystemisperformed.Itoccurseverythreetofivedaysatairportgate.2)Bcheck:Achecktopicsplusinspectionofinternalcontrolsystems,hydraulicsystemsandenergyequipmentisperformed.Itoccurseveryeightmonthsatairportgate.3)Ccheck:Aircraftisopenedupextensivelyandinspectionofwear,corrosionandcracksisperformed.Itoccursevery12to17monthsatthemaintenancehangarofairline’shubairport.4)Dcheck:Aircraftisdisassembledandoverhauledperfectly.Itoccursevery22,500flighthoursatspecializedfacility.AandBchecksarecategorizedastheminorlevelmaintenance.Whereas,CandDchecksareheavymaintenance.Especially,Dcheckaffectsaircraftavailabilityandlifecyclecost.Theairframehastodamage-tolerantduringdesignedoperationperiod.Toassuretheairframedamage-tolerance,thefatiguetestusingfull-scaleairplanestructureisrequiredbyFederalAviationAdministration(FAA)regulation[Dixon(2006);FaderalAviationAdministration(1998)].Theregulationdirectsthatthesufficientfull-scaletestevidencehastobemorethantwotimesoftheprespecifiedoperationintervaltoguaranteetheop-erationsafety.BecausecatastrophictroublesofairframeduringoperationcanbeavoidablewhenappropriateDchecksareundergone,periodsofA,B,andCcheckscanbedeterminedconsideringtheircosts.A,B,C,andDchecksaretheimperfectmaintenanceandsuchmain-tenanceassumesthatcumulativehazardfunctionsafterPMsreduceatacertainrate.ThebasicimperfectPMpolicywasintroduced[Nakagawa(2005)],anditextendedtothethreeechelonsone[ItoandNakagawa(2009)],theNechelonsone[ItoandNakagawa(2010)],thepracticaluse-fulone[ItoandNakagawa(2011)].UtilizingthepracticalusefulimperfectPMpolicy,weconsideredtheoptimaloperationcensoringpolicyofaircraftwhichmaximizestheexpectedprofit[ItoandNakagawa(2012)]. 334September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book320ReliabilityModelingwithApplicationsFig.3Preventivemaintenancescheduleofairframe.Fromthepracticalviewpoint,thereductionrateofcumulativehazardfunctionshouldbedefinedpreciselybecausethesmallchangeofreductionratecausesthegreatchangeofcalculationresults.Inthissection,thetraditionalimperfectmaintenancepolicy[ItoandNakagawa(2009)]anditspracticallyrevicedone[ItoandNakagawa(2011)]ofcivilaircraftarediscussed.4.1Model14.1.1ModelandassumptionsFollowingimperfectpreventivemaintenanceisassumed:1)TheairframeismaintainedpreventivelyattimesiT1(i=1,2,3,···,M1−1),jT2≡jM1T1(j=1,2,3,···,M2−1)andM2T2.M1T1andM2T2aredefinedasT2andS,respectively(seeFig.3).2)CumulativehazardfunctionsH(t)afterPMsreducetoaiH(iT)(0<1a≤1)attimesiTandajM1+ibjH(jT+iT)(0 335September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookMaintenanceModelsofMiscellaneousSystems321Fig.4VarianceofcumulativehazardfunctionwithtimeofModel1.WhenM2=1,thetotalexpectedcostofairframemaintenanceis[2]M∑1−1∫(i+1)T1C(M,1)=caih(t)dt+(M−1)c+c11r11si=0iT1[M∑1−1()]i−1iT2M1−1=cr(1−a)aH+aH(T2)M1i=1+(M1−1)c1+cs.(34)WhenM2=2,thetotalexpectedcostis[M∑1−1∫(i+1)T1M∑1−1∫(i+1)T1]C(M,2)=caih(t)dt+aM1+ibh(T+t)dt11r2i=0iT1i=0iT1+2(M1−1)c1+c2+cs{M∑1−1[()(())]i−1iT2M1i=cr(1−a)aH+abH1+T2M1M1i=1}+a2M1−1bH(2T)+aM1−1(1−ab)H(T)22+2(M1−1)c1+c2+cs.(35) 336September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book322ReliabilityModelingwithApplicationsWhenM2=3,thetotalexpectedcostisC1(M1,3)[M1−1∫(i+1)TM1−1∫(i+1)T∑1∑1=caih(t)dt+aM1+ibh(T+t)dtr2i=0iT1i=0iT1M1−1∫(i+1)T]∑1+a2M1+ib2h(2T+t)dt+3(M−1)c+2c+c2112si=0iT1{M∑1−1[()(())i−1iT2M1i=cr(1−a)aH+abH1+T2M1M1i=1(())]2M12i3M1−12+abH2+T2+abH(3T2)M1}+a2M1−1(1−ab)H(2T)+aM1−1(1−ab)H(T)22+3(M1−1)c1+2c2+cs.(36)From(34),(35)and(36),C1(M1,M2)isM∑2−1M∑1−1∫(i+1)T1C(M,M)=cajM1+ibjh(jT+t)dt112r2j=0i=0iT1+M2(M1−1)c1+(M2−1)c2+csM∑2−1[∑M1(())=cajM1bj(1−a)ai−1Hj+iTrM1j=0i=1]+aM1H((j+1)T)−H(jT)22+M2(M1−1)c1+(M2−1)c2+cs.(37)4.1.2OptimalmaintenancepoliciesThetimeSisfixedbecausethehugecostisnecessaryforitsundergo-ing.OptimalM∗swhichminimizetotalexpectedcostsC(M,M)are1112 337September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookMaintenanceModelsofMiscellaneousSystems323searchedwhenM2=1,2,3,···.FormingtheinequalityC1(M1+1,M2)−C1(M1,M2)≥0,M∑2−1{∑M1[(())M2c1≥ajM1bjai−1Hj+iT2cr(1−a)M1j=0i=1(())]ji−aHj+T2M1+1}1−aj[]−H(jT)−aM1H((j+1)T).(38)221−aSupposethatH(t)=λt,(37)and(38)arerewrittenas,respectively,cλT1−aM11−(aM1b)M2r2C1(M1,M2)=MM11−a1−a1b+M2(M1−1)c1+(M2−1)c2+cs,(39)Mc11−aM11−(aM1b)M221≥crλT2M11−a1−aM1b11−aM1+11−(aM1+1b)M2−.(40)M1+11−a1−aM1+1bWhenM2=1,(39)and(40)arerewrittenas,respectively,cλT1−aM1r2C1(M1,1)=+(M1−1)c1+cs,(41)M11−a∑M1c11i−1≥ia.(42)(1−a)λT2c0M1(M1+1)i=1BylettingL(M1)betheright-handsideof(42),1L1(1)=,limL1(M1)=0,(43)2M1→∞L1(M1)−L1(M1+1)∑M1=2iai−1(1−aM1+i−1)>0.(44)M1(M1+1)(M1+2)i=1Therefore,L1(M1)decreaseswithM1andwehavethefollowingoptimalpolicywhenM2=1:1)Ifc/((1−a)λTc)≥1/2,thenM∗=1.12r12)Ifc/((1−a)λTc)<1/2,thenthereexistsafiniteanduniqueM∗(1<12r1M∗<∞).1 338September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book324ReliabilityModelingwithApplicationsFig.5Anexampleoffuselageskincrosssection.4.2Model2ForapplyingModel1,parametersaandbhavetobedeterminedanditmaycreateconfusioninpracticebecausetheslightchangeofaandbcausethegreatchangeofPMpolicyandhowtodetermeaandbwithhighaccuracyisaproblematicissue.Fromthepracticalviewpoint,settlingsuchsensibleparametersshouldbeeliminated.Figure5denotesanexampleoffuselageskincrosssectionandwecaneasilyfindthatstringerscannotbeinspectedexceptthattheinnerpanelortheouterskinisstrippedoffatthemaintenancehangarandspecializedfacility.Whenthetotalcumulativehazardfunctionisconstitutedwithdis-cretecumulativehazardfunctionsofinnerpanel,outerskinandstringers,cumulativehazardfunctionsofinnerpanelandouterskincanberenewedatAandBchecksandthestringerhazardfunctioncanberenewedatCandDchecks.Usingabovesheme,Model1canbeexpandedasfollows.4.2.1ModelandassumptionsWeconsiderthefollowingmaintenancepolicywhichhasthesameassump-tionsexcept2):2’)IfthePMisnotmade,thecumulativehazardfunctionofairframeisconstitutedwithfourparts,i.e.,H(t)=H0(t)+H1(t)+H2(t)+Hs(t),(45)whereH1(t)becomesnewattimesiT1becauseallfailuresofthispartarediscoveredandarerepaired,H2(t)becomesnewattimesjT2 339September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookMaintenanceModelsofMiscellaneousSystems325becauseallfailuresofthispartarediscoveredandarerepaired,Hs(t)becomesnewattimeSbecausethesystemisopenedup,inspectedandrepairedextensively,andH0(t)isunchangedatanymaintenanceandoverhaul.CumulativehazardfunctionsH(t)before(−)andafter(+)PMattimeiT1aredenotedasfollows,respectively.H−(iT1)=H1(T1)+H2(iT1)+Hs(iT1)+H0(iT1),H+(iT1)=H2(iT1)+Hs(iT1)+H0(iT1),(46)andH(t)before(−)andafter(+)PMattimejT2aredenotedasfollows,respectively.H−(jT2)=H1(T1)+H2(T2)+Hs(jT2)+H0(jT2),H+(jT2)=Hs(jT2)+H0(jT2).(47)Similarly,H(S)before(−)andafter(+)overhaularedenotedasfollows,respectively,H−(S)=H1(T1)+H2(T2)+Hs(S)+H0(S),H+(S)=H0(S).(48)Theexpectednumberoffailuresin[0,S]isM2−1{M1−1[∫(i+1)T+jT]}∑∑12h(t)dt−H1(T1)−H2(T2)j=0i=0iT1+jT2=H(S)−M2M1H1(T1)−M2H2(M1T1)≡H012(S)−M2M1H1(T1)−M2H2(M1T1),(49)whereH012(t)≡H0(t)+H1(t)+H2(t)+Hs(t).Theexpectedcostrateofairframemaintenancein[0,S]is{[1C2(M1,M2)=crH012(M2M1T1)−M2M1H1(T1)M2M1T1}]−M2H2(M1T1)+c1M2M1+c2M2+cs.(50) 340September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book326ReliabilityModelingwithApplications4.2.2OptimalmaintenancepoliciesWefindoptimalM∗andM∗whichminimizeC(M,M)in(50).Forming12212theinequalityC2(M1,M2+1)−C2(M1,M2)≥0,[]H012((M2+1)M1T1)H012(M2M1T1)csM2(M2+1)M1T1−≥.(M2+1)M1T1M2M1T1cr(51)Supposethath012(t)andh2(t)arestrictlyincreasingfunctionswithtandhi(∞)=∞.Denotingtheleft-handsideof(51)byL21(M2),L21(M2)isalwayspositive,L21(∞)=∞andL21(M2)isstrictlyincreasingfunctionwithM2.Therefore,wehavethefollowingoptimalpolicy:1)IfL(1)≥c/c,thenM∗=1.21sr22)IfL21(1) 341September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookMaintenanceModelsofMiscellaneousSystems327whereL3(n,t)≡[H(nt)−nH(t)]/nt.Whenh(t)isastrictlyincreasingfunction,H(nt)−nH(t)L3(n,t)=nt[∫nt∫t]1=h(u)du−nh(u)dunt00n[∫it∫t]1∑=h(u)du−h(u)du>0,(54)nti=1(i−1)t0H(nt)−nH(t)L3(n,∞)=limt→∞nt[]H(nt)H(t)=lim−t→∞ntt=lim[h(nt)−h(t)]=∞,(55)t→∞dL3(n,t)[h(nt)n−nh(t)]t−[H(nt)−nH(t)]=dtnt21=[nth(nt)−H(nt)−nth(t)+nH(t)]nt2n{∫jt∫t}1∑=[h(nt)−h(u)]du−[h(t)−h(u)]dunt2j=1(j−1)t0∑n∫t1={h(nt)−h[u+(j−1)t]−h(t)+h(u)}du>0,nt2j=10(56)forn>1.Thus,L23(M1)isalwayspositive.Furthermore,L23(M1+1)−L23(M1){∫∫M2(M1+2)T1M2(M1+1)T1=(M1+1)h2(u)du−h2(u)duM2(M1+1)T1M2M1T1[∫∫]}(M1+2)T1(M1+1)T1−M2h2(u)du−h2(u)du.(57)(M1+1)T1M1T1 342September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book328ReliabilityModelingwithApplicationsThus,L23(M1)isstrictlyincreasingfunctionwithM1when(57)ispositive.Therefore,wehavethefollowingoptimalpolicywhenL22(M1)isdenotedbytheleft-handsideof(52):1)IfL(1)≥(cM+c)/c,thenM∗=1.2222sr12)IfL22(1)<(c2M2+cs)/cr,thenthereexistsafiniteanduniquemini-mumM∗(1 343September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookMaintenanceModelsofMiscellaneousSystems329ReferencesBarlow,R.E.andProschan,F.(1965).MathematicalTheoryofReliability(JohnWiley&Sons).Bogdanoff,J.L.andKozin,F.(1985).ProbabilisticModelsofCumulativeDamage(JohnWiley&Sons).C¸inlar,E.(1975).IntroductiontoStochasticProcesses(Prentice-Hall).Cox,D.R.(1962).RenewalTheory(Methuen).Dixon,M.(2006).ThemaintenanceCostsofAgingAircract:InsightsfromCom-mercialAviation(RANDCorporation).Esary,J.D.,Marshall,A.W.andProschan,F.(1973).Shockmodelsandwearprocesses,AnnalsofProbability,Vol.1,pp.627–649.FaderalAviationAdministration(1998).FatigueEvaluationofStructure;FinalRule.March31,63,61:15708-15715.(14CFRPart25,DocketNo.27358,AmendmentNo.25-96).FAAAdvisoryCircular(1998).DamageToleranceandFatigueEvaluationofStructure.AC25,571-1C.Hisano,K.(2000).PreventiveMaintenanceandResidualLifeEvaluationTech-niqueforPowerPlant(I.PreventiveMaintenance)(inJapanese),TheTher-malandNuclearPower,Vol.51,No.4,pp.491–517.Hisano,K.(2001).PreventiveMaintenanceandResidualLifeEvaluationTech-niqueforPowerPlant(V.ReviewofFutureAdvancesinPreventiveMainte-nanceTechnology)(inJapanese),TheThermalandNuclearPowerVol.52,No.3,pp.363–370.Ito,K.,Nakagawa,T.andTeramoto,K.(2006).OptimalMaintenancePolicyforaSystemwithDamageRepair,Proceedingsofthe2ndAsianInternationalWorkshop(AIWARM2006)AdvancedReliabilityModelingII,ReliabilityTestingandImprovement,Busan,Korea,pp.235–242.Ito,K.andNakagawa,T.(2006).OptimalOperationCensoringPolicyforaDam-agedSystem,ProceedingsofInternationalWorkshoponRecentAdvancesinStochasticOperationsResearchII(RASORNanzan),Nagoya,Japan,pp.106–111.Ito,K.andNakagawa,T.(2007).OptimalOperationCensoringPolicyforaSys-temwithDamageRepair,Proceedingsofthe13thISSATInternationalCon-ferenceonReliabilityandQualityinDesign,Seattle,Washington,U.S.A,pp.216–220.Ito,K.andNakagawa,T.(2008).ComparisonofThreeCumulativeDamageMod-els,Proceedingsofthe3rdAsianInternationalWorkshop(AIWARM2008),Taichung,Taiwan,pp.332–338.Ito,K.andNakagawa,T.(2009).OptimalImperfectMaintenanceofAircraft,Pro-ceedingsofthe15thISSATInternationalConferenceReliabilityandQualityinDesign,SanFrancisco,California,U.S.A.,pp.215–218.Ito,K.andNakagawa,T.(2010).OptimalMulti-echelonMaintenanceofAircraft,Proceedingsofthe4thAsia-PacificInternationalSymposium(APARM 344September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book330ReliabilityModelingwithApplications2010),AdvancedReliabilityModelingIV,Wellington,NewZealand,pp.257–264.Ito,K.andNakagawa,T.(2011).OptimalMaintenanceofAircraft,Proceed-ingsofThe7thInternationalConferenceon“MathematicalMethodsinReliability”:Theory.Methods.Applications.(MMR2011),Beijing,China,pp.859–864.Ito,K.andNakagawa,T.(2012).OptimalOperationCensoringPolicyofAircraft,ProceedingsoftheAsia-PacificInternationalSymposium(APARM2012),Nanjing,China,pp.184–191.Kosugiyama,S.,Takizuka,T.,Kunitomi,K.,Yan,X.,Katanishi,S.andTakada,S.(2003).BasicPolicyofMaintenanceforthePowerConversionSys-temoftheGasTurbineHighTemperatureReactor300(GTHTR300)(inJapanese),JournalofNuclearScienceandTechnology,Vol.2,No.3,pp.105–117.Nakagawa,T.andOsaki,S.(1974).Someaspectsofdamagemodel,Microelec-tronicsandReliability,Vol.13,pp.253–257.Nakagawa,T.(2005).MaintenanceTheoryofReliability(Springer-Verlag).Nakagawa,T.(2007).ShockandDamageModelsinReliabilityTheory(SpringerVerlag).Nakagawa,T.andIto,K.(2008).OptimalMaintenancePoliciesforaSystemWithMultiechelonRisks,IEEETransactionsonSystems,Man,andCyber-netics,PartA:SystemsandHumansVol.38,pp.461–469.Osaki,S.(1992).AppliedStochasticSystemsModeling(SpringerVerlag).Paul,D.,Kelly,L.andVenkayya,V.(2002).EvolutionofU.S.MilitaryAircraftStructuresTechnology,JournalofAircraft,Vol.39,No.1,pp.18–29.Ross,S.M.(1983).StochasticProcesses(JohnWiley&Sons). 345September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookChapter18ReliabilityAnalysisofaSystemConnectedwiththeRadioLinkMitsuhiroImaizumiCollegeofContemporaryManagement,AichiGakusenUniversity,1Shiotori,Ohike-cho,Toyota471-8532,Japan1IntroductionAsmobiledeviceshavebeenwidelyusedandradiocommunicationtech-nologyhasremarkablydeveloped,thedemandforimprovementofthereli-abilityofradiocommunicationhasgreatlyincreased.Generallycomparedwiththewiredlink,packetlossoccurswithhighprobabilityontheradiolinkduetoradiointerferencesuchasphasing[Miyoshi,SuganoandMu-rata(2002);Yuki,Yamamoto,Sugano,Murata,MiyaharaandHatauchi(2002);Takahashi,Saito,Aida,TobeandTokuda(2005);Itaya,Hasegawa,Hasegawa,Davis,KadowakiandObana(2005)].Therefore,howweim-provethethroughputefficiencyontheradiolinkhasbecomeaproblem.Inthischapter,wepayattentiontoasystemconnectedwiththeradiolink,andtreatthreestochasticmodels.UsingthetheoryofMarkovrenewalpro-cesses,wederivethemeasuressuchasthemeantimeuntilthetransmissionofpacketssucceeds.ThetheoryofMarkovrenewalprocesses[Osaki(1992)]isusedtoana-lyzethesystem.BarlowandProschan[BarlowandProschan(1965)]gaveatableofapplicablestochasticprocessassociatedrepairmanproblems.Nak-agawaandOsaki[NakagawaandOsaki(1979)]analyzedtwo-unitsystemsusingauniquemodificationoftheregenerationpointtechniquesofMarkovrenewalprocesses.InSection2and3,weformulateastochasticmodelofanetworksys-temwithretransmissioncontrolschemeusingduplicatedACK(positive331 346September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book332ReliabilityModelingwithApplicationsacknowledgment)packets.IntheTCPprotocol,thehighreliabilityofthedatatransmissionisrealizedbyreturninganACKpacketfromareceiverwhenasendertransmitsdatapackets,butontheradiolink,itisnotraretoloseanACKpacket.Inordertocopewiththisproblem,severalschemeshavebeenconsidered[MatsudaandYamamoto(2004);Zhang,ShiraziandTanaka(2004);SaitoandTeraoka(2004);BalakrishnanandKatz(1998);BakreandBadrinath(1995)].Asoneofschemestocopewiththisproblem,theretransmissionschemeusingduplicatedACKpacketswhereasenderreturnsredundantACKpacketshasbeenproposed[Miyoshi,SuganoandMurata(2001)].InSection4,weformulateastochasticmodelofanetworksystemwithselectiverepeatdatatransmissionerrorrecoveryscheme.InSection2and3,weassumethatifasenderhasfailedtotransmitdatapackets,itretransmitsallpackets.Ontheotherhand,inSection4,weassumethatifasenderhasfailedtotransmitdatapackets,itretransmitspacketsthatthesenderhasfailedtotransmit.Althoughthesimulationaboutthepolicyforthesystemconnectedwiththeradiolinkhasalreadybeenintroduced,therearefewformalizedstochas-ticmodels.IneachModel,themeantimeuntilthetransmissionsucceedsisana-lyticallyderived.Further,anoptimalpolicywhichmaximizestheamountofdatatransmissionsperunitoftimeuntilthetransmissionsucceedsisdiscussed.2Model12.1ModelandAnalysisWepayattentiontoonlyadatatransmissionterminalconnectedtothenetworkwiththeradiolink.TheDataistransmittedfromthefixedtermi-naltothemobileterminalviathebasestation.Fig.1drawstheoutlineofthemodel.(1)Asendertransmitstherequirementforconnectionestablishmenttoareceiver.Afterreceivingtherequirementforconnectionestablishment,thereceiverreturnsaresponse.TheconnectionestablishmentneedsthetimeaccordingtoageneraldistributionA(t)withfinitemeana.(2)Thesendertransmitsndatapacketstothereceiverwherethecon-nectionisestablished.Theeditingtimeforeachdatapacketneeds 347September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisofaSystemConnectedwiththeRadioLink333DatapacketsACKpacketsBasestationFixedstationMobilestationFig.1Outlineofthemodel.thetimeaccordingtoageneraldistributionB1(t)withfinitemeanb1.ThetransmissionfromthesendertothereceiverneedsaccordingtoageneraldistributionD1(t).(3)Afterthereceiverhavereceivedndatapackets,itreturnanACK(posi-tiveacknowledgment)packetifithasreceivedalldatapacketscorrectly.TheeditingtimeforeachACKpacketneedsthetimeaccordingtoageneraldistributionB2(t)withfinitemeanb2.IfthesenderhasnotreceivedanACKpacketandithasjudgedtimeout,itretransmitsndatapackets.ThetimetotransmitanACKpackethasageneraldistributionD2(t),anddenotesthatD(t)≡D1(t)∗D2(t)withfi-nitemeand.TheasteriskmarkdenotestheStieltjesconvolutionand∫tΦ1(t)∗Φ2(t)≡0Φ2(t−u)dΦ1(u).WeassumethatthetimeuntilthesenderjudgestimeouthasthesamedistributionD(t),theprobabilitythatadatapacketlossoccursispandtheprobabilitythatanACKpacketlossoccursisq.(4)Whenthereceiverhasreceivedretransmissiondatapacket(thesec-ondtransmissionpacket),itreturns2ACKpackets.ThetransmissionsucceedsifthesenderhasreceivedatleastanACKpacket.WhenthesenderhasjudgedtimeoutwithoutreceivingallACKpackets,itretransmitsnpackets.Generally,whenthereceiverhasreceivedthe(k−1)-thtimeretransmissiondatapacket(thek-thtimetransmissionpacket),itreturnskACKpackets,andwhenthesenderhasjudgedtimeoutwithoutreceivingallACKpackets,itretransmitsndatapackets.(5)Ifthetransmissionhavefailedktimes,oncethesenderinterruptstheretransmission,andrestartsagainfromthebeginningafteraconstanttimegwhereG(t)≡0fort 348September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book334ReliabilityModelingwithApplicationsF01SFig.2Transitiondiagrambetweensystemstates.StateF:Retransmissionfailsk(k=1,2,···)timesandisinterrupted.StateS:Transmissionofnpacketssucceeds.ThesystemstatesdefinedaboveformaMarkovrenewalprocesswherestateSisanabsorbingstate.TransitiondiagrambetweensystemstatesisshowninFig.2.LetQij(t)(i=0,1,F;j=0,1,S,F)beone-steptransitionprobabili-tiesofaMarkovrenewalprocess.Then,bythesimilarmethodof[Yasui,NakagawaandSandoh(2002)],Q01(t)=A(t),(n)Q1S(t)=γ1[B1(t)∗B2(t)∗D(t)](n)(n)(2)+γ2[B1(t)∗B2(t)∗D(t)]∗[B1(t)∗B2(t)∗D(t)]+···(n)(n)(2)+γk[B1(t)∗B2(t)∗D(t)]∗[B1(t)∗B2(t)∗D(t)](n)(k)∗···∗[B1(t)∗B2(t)∗D(t)],()∑k(n)Q1F(t)=1−γi[B1(t)∗B2(t)∗D(t)]i=1(n)(2)(n)(k)∗[B1(t)∗B2(t)∗D(t)]∗···∗[B1(t)∗B2(t)∗D(t)],QF0(t)=G(t),(1)whereΦ(i)(t)denotesthei-foldStieltjesconvolutionofadistribution∫tΦ(t)withitself,i.e.,Φ(i)(t)≡Φ(i−1)(t)∗Φ(t),Φ(t)∗Φ(t)≡Φ(t−1202u)dΦ(u),Φ(0)(t)≡1.TheprobabilitythatasenderisreceivedACKpack-1etsatk-thtransmissionwithoutreceivingACKpacketsat(k−1)-thtrans-missionis()k∑−1γ≡1−γ(1−p)n(1−qk)(k=1,2,···),(2)kii=1∑0where≡0.i=1 349September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisofaSystemConnectedwiththeRadioLink335Letϕ(s)betheLaplace-Stieltjes(LS)transformofanyfunctionΦ(t),∫∞i.e.,ϕ(s)≡e−stdΦ(t)forRe(s)>0.Then,from(1),0q01(s)=a(s),∑ki(i+1)niq1S(s)=γi[b1(s)d(s)]b2(s)2,i=1()∑kk(k+1)nkq1F(s)=1−γi[b1(s)d(s)]b2(s)2,i=1qF0(s)=g(s).(3)Wederivethemeantimeℓ1(n)untilthetransmissionofnpacketssuc-ceeds.LetHiS(t)(i=0,1,F)bethefirst-passagetimedistributionfromstateitostateS.Then,wehavethefollowingrenewalequations:H0S(t)=Q01(t)∗H1S(t),H1S(t)=Q1F(t)∗HFS(t)+Q1S(t),HFS(t)=QF0(t)∗H0S(t).(4)TakingtheLStransformonthebothsidesof(4)andsolvingrenewalequations,q01(s)q1S(s)h0S(s)=.(5)1−q01(s)q1F(s)qF0(s)Hence,themeantimeℓ1(n)untilthetransmissionsucceedsisgivenby∫∞−d[h0S(s)]ℓ1(n)≡tdH0S(t)=lim0s→0dsk∑−1[]i(i+1)γii(nb1+d)+b22i=1k∑−1[]k(k+1)+(1−γi)k(nb1+d)+b2+a+g2i=1=−g.(6)∑kγii=1Whenk=1,ℓ1(n)isgivenbynb1+b2+d+a+gℓ1(n)=−g.(7)(1−p)n(1−q) 350September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book336ReliabilityModelingwithApplications2.2OptimalPolicyWediscussanoptimalpolicywhichmaximizesthetransmissionofallpack-etsperunitoftimeuntilthetransmissionsucceeds.Wedefinethethrough-putE1(n),whichrepresentstherateofnpacketstotheirmeantransmissiontimes,asthefollowingequation:nE1(n)≡ℓ1(n)∑knγii=1=k−1[]∑i(i+1)γii(nb1+d)+b22i=1()k∑−1[]∑kk(k+1)+1−γik(nb1+d)+2b2+a+g−gγii=1i=1(n=1,2,···).(8)Weseekanoptimaln∗whichmaximizesE(n).Weputformallythat11V(n)≡1/E(n)andseekn∗whichminimizesV(n).Fromtheinequality1111V1(n+1)−V1(n)≥0,nX1(n+1)−(n+1)X1(n)+g≥0,(9)wherek∑−1[]i(i+1)γii(nb1+d)+b22i=1()k∑−1[]k(k+1)+1−γik(nb1+d)+2b2+a+gi=1X1(n)≡.∑kγii=1Denotingtheleft-handsideofequation(9)byL1(n),L1(n+1)−L1(n)=(n+1)J1(n),(10)whereJ1(n)≡[X1(n+2)−X1(n+1)]−[X1(n+1)−X1(n)].Hence,whenX1(n)isaconvexfunctionandJ1(1)>0,L1(n)isstrictlyincreasinginnfromL1(1)toL1(∞). 351September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisofaSystemConnectedwiththeRadioLink337Therefore,whenJ1(1)>0,wehavethefollowingoptimalpolicy:(1)IfL1(1)<0andL1(∞)>0thenthereexistsafiniteanduniquen∗(>1)whichsatisfies(9).1(2)IfL(∞)≤0thenn∗=∞.11(3)IfL(1)≥0thenn∗=1.113Model23.1ModelandAnalysisWemodifyModel1.Weassumethatwhenthereceiverhasreceivedthe(k−1)-thtimeretransmissiondatapacket(thek-thtimetransmissionpacket),itreturnsm(1≤m<∞)ACKpackets.Thatis,weassumethatthenumberofACKpacketsdoesnotdependsonk.Then,theLStransformsofone-steptransitionprobabilitiesQij(t)aregivenbythefollowingequations:q01(s)=a(s),q(s)=(1−p)n(1−qm)[b(s)nb(s)md(s)]1S12+[1−(1−p)n(1−qm)]×(1−p)n(1−qm)[b(s)nb(s)md(s)]212+[1−(1−p)n(1−qm)]2×(1−p)n(1−qm)[b(s)nb(s)md(s)]312+···+[1−(1−p)n(1−qm)]k−1×(1−p)n(1−qm)[b(s)nb(s)md(s)]k,12q(s)=[1−(1−p)n(1−qm)]k[b(s)nb(s)md(s)]k,1F12qF0(s)=g(s).(11)Then,bythesimilaranalysisofModel1,wehavethemeantimeℓ2(n)untilthetransmissionsucceedsasfollows:∑k[1−(1−p)n(1−qm)]i−1i=1×(1−p)n(1−qm)i(nb+mb+d)12+[1−(1−p)n(1−qm)]k[k(nb+mb+d)+g]+a12ℓ2(n)=.(12)1−[1−(1−p)n(1−qm)]k 352September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book338ReliabilityModelingwithApplications3.2OptimalPolicyIntermsofModel2,wedefinethethroughputE2(n),whichrepresentstherateofnpacketstotheirmeantransmissiontimes,asthefollowingequation:nE2(n)≡ℓ2(n)n{1−[1−(1−p)n(1−qm)]k}=∑k[1−(1−p)n(1−qm)]i−1i=1×(1−p)n(1−qm)i(nb+mb+d)12+[1−(1−p)n(1−qm)]k[k(nb+mb+d)+g]+a12(n=1,2,···).(13)Weseekanoptimaln∗whichmaximizesE(n).Bysimilaranalysisof22Model1,wecandiscusstheoptimalpolicyforModel2.4Model34.1ModelandAnalysisWemodifyModel1.Model1and2areGo-Back-Ntypemodel.Ontheotherhand,Model3isSelective-Repeattypemodel.Wehavethefollowingassumptions.Afterthereceiverhavereceivedndatapackets,itreturnsanACKpacketoraNAK(negativeacknowledgment)packetbydependingonwhetheralldatapacketshavereceivedcorrectlyornot.TheeditingtimeforeachACKorNAKpacketneedsthetimeaccordingtoageneraldistri-butionB2(t)withfinitemeanb2.IfthesenderhasreceivedaNAKpacket,itretransmitsdatapacketsthatthesenderhasfailedtotransmit.IfthesenderhasfailedtoreceiveanACKpacketoraNAKpacket,thenitisjudgedtimeoutanditretransmitsndatapackets.ThetimetotransmitanACKpacketoraNAKpackethasageneraldistributionD2(t),anddenotesthatD(t)≡D1(t)∗D2(t)withfinitemeand.WeassumethatthetimeuntilthesenderjudgestimeouthasthesamedistributionD(t)andtheprobabilitythatadatapacketlossoccursisp.Particularly,intermsofModel3,theprobabilitythatanACKpacketlossoraNAKpacketlossoccursisq=0. 353September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisofaSystemConnectedwiththeRadioLink339Then,theLStransformsofone-steptransitionprobabilitiesQij(t)aregivenbythefollowingequations:q01(s)=a(s),q(s)=(1−p)nb(s)n[b(s)d(s)]1S12n()∑n+pm1(1−p)nb(s)n+m1[b(s)d(s)]212m1m1=1∑n(n)∑m1(m)+1pm1+m2(1−p)nm1m2m1=1m2=1×b(s)n+m1+m2[b(s)d(s)]312+···∑n()∑m1()m∑k−2()nm1mk−2+···m1m2mk−1m1=1m2=1mk−1=1×pm1+m2+···+mk−1(1−p)n×b(s)n+m1+m2+···+mk−1[b(s)d(s)]k,12∑n()∑m1()m∑k−2()nm1mk−2q1F(s)=···m1m2mk−1m1=1m2=1mk−1=1×pm1+m2+···+mk−1(1−p)n−mk−1×b(s)n+m1+m2+···+mk−2[b(s)d(s)]k−112×[1−(1−p)mk−1]b(s)mk−1[b(s)d(s)],12qF0(s)=g(s).(14)Itisdifficulttoderiveexplicitlythemeantimeℓ3(n)untilndataunittransmissionssucceed.Weconsidertheparticularcasek=2i.e.,iftheretransmissionhasfailed2times,thesenderinterruptstheretransmission.Inthiscase,fromequation(5),∫∞−d[h0S(s)]ℓ3(n)≡tdH0S(t)=lim0s→0dsa+nb(1+p)+(b+d)[2−(1−p)n]+112µ1=−.(15)(1−p2)nµ 354September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book340ReliabilityModelingwithApplicationsSimilarly,themeantimewhenk=3,i.e.,iftheretransmissionhasfailed3times,thesenderinterruptstheretransmission,isgivenbya+nb(1+p+p2)1[]∑2+(b+d)3−(1−pi)n+12µi−11ℓ3(n)=−.(1−p3)nµMoreover,fromtheseanalyses,themeantimeforanyk≥1isa+nb[1+p(1−pk−2)+(k−1)pk−1]1[]k∑−1+(b+d)k−(1−pi)n+12µi=11ℓ3(n)=−.(16)(1−pk)nµ4.2OptimalPolicyWediscussanoptimalpolicywhichmaximizesthethroughputE3(n)intermsofModel3.WedefinethethroughputE3(n),whichpresentstheratioofndataunitstotheirmeantransmissiontimes,asthefollowingequation:nE3(n)≡ℓ3(n)n(1−pk)n=a+nb[1+p(1−pk−2)+(k−1)pk−1]1[]k∑−1in11kn+(b2+d)k−(1−p)+−(1−p)µµi=1(n=1,2,···).(17)Weseekanoptimalnumbern∗whichmaximizesE(n).Weputformally33thatV(n)≡1/E(n)andseekn∗whichminimizesV(n).Fromthein-3333equalityV3(n+1)−V3(n)≥0,nℓ3(n+1)−(n+1)ℓ3(n)≥0.(18) 355September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisofaSystemConnectedwiththeRadioLink341Denotingtheleftsideof(18)byL3(n),n+1L3(n+1)−L3(n)=(1−p2)n+2J3(n),[]a+1+k(b+d)(−1+2pk)µ2+2bpk[1+p(1−pk−2)+(k−1)pk−1]1[]k∑−1−(b+d)(1−pi)(−1+2pk−pi)2i=11L3(1)=+,(1−pk)2µL3(∞)=∞,whereJ(n)≡ap2k+bpk(2+npk)[1+p(1−pk−2)+(k−1)pk−1]31[]k∑−12kinki212k+(b2+d)kp−(1−p)(p−p)+p,µi=1whichisstrictlyincreasinginn.Thatis,whenJ3(1)>0,L3(n)isstrictlyincreasinginnfromL3(1)to∞.Therefore,whenJ3(1)>0,wehavethefollowingoptimalpolicy:(i)IfL(1)<0,thenthereexistsafiniteanduniquen∗(>1)which33satisfies(18).(ii)IfL(1)≥0thenn∗=1.335NumericalExampleWecomputenumericallytheoptimalnumberwhichmaximizesthethrough-put.Supposethatthemeantimeaforconnectionestablishmentisaunittimeinordertotherelativetendencyofperformancemeasures.ItisassumedthatthemeaneditingtimeforanACKpacketoraNAKpacketisb/a=1(×10−1),themeaneditingtimeforadatapacketis2b1=2b2,themeantimetotransmitadatapacketandanACKpacketoraNAKpacketisd/a=1∼4(×10−1),themeantimefromthesenderinterruptsretransmissiontorestartagainisg/a=2,theprobabilitythatadatapacketlossoccursisp=0.01∼0.3andtheprobabilitythatanACKpacketlossoccursisq=0.01∼0.3. 356September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book342ReliabilityModelingwithApplicationsTable1OptimalnumberntomaximizeE(n)andE(n)whenk=2.11d/apq(×10 1)0.010.050.10.20.30.0121212121213.1733.0752.9532.7102.4550.05888881.9211.8651.7941.6461.48610.1555551.3471.3091.2591.1541.0390.2333330.8360.8120.7820.7160.6440.3222220.5930.5770.5550.5080.4560.0122222222223.1153.0192.8992.6602.4100.05888881.8531.8001.7311.5881.43420.1555551.2881.2511.2041.1030.9940.2333330.7920.7700.7410.6790.6110.3222220.5590.5440.5230.4790.4310.0123232323233.0102.9162.8012.5702.3280.05888881.7321.6811.6171.4841.34040.1555551.1841.1501.1071.0150.9150.2333330.7180.6980.6720.6160.5550.3222220.5020.4880.4690.4300.387Table1givestheoptimalnumbern∗whichmaximizesthethroughput1andE(n∗)whenk=2.Forexample,whend/a=0.1,p=0.05and11q=0.01,theoptimalnumberisn∗=8andE(8)=1.921.Thisindicates11thatn∗decreaseswithp,however,increaseswithd/a.Table1alsoindicates1thatn∗dependslittleonq.1Next,intermsofthethroughput,wecompareModel1withModel2.Table2givesthethroughputE(n∗)whenk=3andd/a=0.1incaseof11Model1.Table3givesthethroughputE(n∗)whenk=3,m=3and22d/a=0.1incaseofModel2.TheseindicatethatthethroughputofModel1islargerthanthatofModel2.ThedifferencebetweenE(n∗)andE(n∗)decreaseswithpand1122q.Especially,whenpissmall,thedifferenceofthroughputislarge. 357September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisofaSystemConnectedwiththeRadioLink343Table2ThroughputE1(n)whenk=3inthecaseofModel1.1pq0.010.050.10.20.30.017.3226.5545.7754.6313.8070.053.0342.8792.7012.3822.0950.11.9531.8761.7841.6101.4420.21.1441.1111.0690.9840.8940.30.8070.7850.7580.7020.641Table3ThroughputE2(n)whenk=3andm=3inthecaseofModel2.2pq0.010.050.10.20.30.013.1393.1393.1363.1163.0620.051.9351.9351.9341.9211.8880.11.3921.3921.3911.3811.3560.20.8960.8960.8950.8890.8720.30.6470.6470.6460.6420.629Table4ThroughputE2(n)whenm=1andk=3inthecaseofModel2.2pq0.010.050.10.20.30.013.2243.1082.9582.6502.3280.052.0691.9921.8941.6891.4780.11.5221.4631.3881.2331.0720.21.0040.9630.9110.8040.6960.30.7380.7070.6680.5880.508Table4givesthethroughputE(n∗)whenm=1,k=3andd/a=0.122inthecaseofModel2.ThisindicatesthatthethroughputofTable4issmallerthanthatofTable2.ComparedwithTable3,ThethroughputofTable4islargerthanthatofTable3underthesamevalueofpwhenqissmall,conversely,itissmallerthanthatofTable3whenqislarge.Table5OptimalnumberntomaximizeE3(n)whenk=2.3pd/a(×10 1)1240.011515160.055550.13330.2222 358September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book344ReliabilityModelingwithApplicationsTable5givestheoptimalnumbern∗whichmaximizesthethroughput3whenk=2andd/a=0.1inthecaseofModel3.Thisindicatesthatn∗3decreaseswithp,however,increaseswithd/a.6ConclusionWehaveformulatedthreestochasticmodelsofasystemconnectedwiththeradiolink,andhavediscussedtheoptimalpolicywhichmaximizesthethroughputuntilthetransmissionsucceeds.Moreover,wehavegivennumericalexamplesandhaveevaluatedthemforvariousstandardparameters.Fromnumericalexamples,wehaveshownthattheoptimalnumberwhichmaximizesthethroughputdecreaseswiththetimetotransmitapacketandtheprobabilitythatapacketlossoccursanddependslittleontheprobabilitythatanACKpacketlossoccurs.Ifsomeparametersareestimatedfromactualdata,wecouldselectthebestpolicy.Itwouldbeveryimportanttoevaluateandimprovethereliabilityofanetworkconnectedwiththeradiolink.Theresultsderivedinthispaperwouldbeappliedinpracticalfieldsbymakingsomesuitablemodificationandextensions.Furtherstudiesforsuchsubjectwouldbeexpected.ReferencesBakre,A.andBadrinath,B.R.(1995).I-TCP:IndirectTCPformobilehosts,Proc.15thICDCS,pp.136–143.Balakrishnan,H.andKatz,R.H.(1998).Explicitlossnotificationandwirelesswebperformance,Proc.IEEEGlobecomInternetMini-Conference.Barlow,R.E.andProschan,F.(1965).MathematicalTheoryofReliability,Wiley,NewYork.Itaya,S.,Hasegawa,J.,Hasegawa,A.,Davis,P.,Kadowaki,N.andObana,S.(2005).StabilizationofLargeAd-hocWirelessNetworksinUnstableRadioEnvironments,IPSTrans.46,pp.2848–2856.Matsuda,T.andYamamoto,M.(2004).RecentResearchActivitiesinWirelessTCPIEICEJ.87,pp.589–594.Miyoshi,M.,Sugano,M.andMurata,M.(2001).AStudyforImprovingTCPPerformanceonRadioTerminal,IEICEJapan,MobileCommunicationWorkshop.Miyoshi,M.,Sugano,M.andMurata,M.(2002).PerformanceEvaluationofTCPThroughputwithConsiderationonLowerLayerProtocolsforWirelessCellularNetworks,IEICETrans.J85-B,pp.732–743. 359September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookReliabilityAnalysisofaSystemConnectedwiththeRadioLink345Nakagawa,T.andOsaki,S.(1979).BibliographyforAvailabilityofStochasticSystems,IEEETrans.Reliability,R-25,4,pp.284–287.Osaki,S.(1992).Appliedstochasticsystemmodeling(Springer,Berlin).Saito,S.andTeraoka,F.(2004).Implementation,AnalysisandEvaluationofTCP-J:ANewVersionofTCPforWirelessNetworks,IPSTrans.J87-D-I,pp.508–515.Takahashi,H.,Saito,M.,Aida,H.,Tobe,Y.andTokuda,H.(2005).RealEnviron-mentEvaluationsofaRoutingSchemeBasedonEstimatedTCPThrough-putforMANET,IPSTrans.46,pp.2857–2870.Yasui,K.,Nakagawa,T.andSandoh,H.(2002).Reliabilitymodelsindatacom-municationsystems,inStochasticModelsinReliabilityandMaintenance,ed.Osaki,S.pp.281–301(Springer,Berlin).Yuki,T.,Yamamoto,T.,Sugano,M.,Murata,M.,Miyahara,H.andHatauchi,T.(2002).AStudyonPerformanceImprovementofTCPoveranAdHocNetwork,IEICETrans.J85-B,pp.2045–2053.Zhang,B.,Shirazi,M.N.andTanaka,S.(2004).ImprovingWirelessTCPPerformancewithExplicitWirelessLossNotificationUsingMAC-layerInformation,IPSTrans.45,pp.1234–1244. 360May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank 361September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookPART5StudiesonReliabilityandMaintenance347 362May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank 363September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookChapter19StudiesonReliabilityandMaintenanceToshioNakagawaDepartmentofBusinessAdministration,AichiInstituteofTechnology,1247Yachigusa,Yakusa-cho,Toyota470-0392Japan1IntroductionIhavemainlystudiedreliabilityandmaintenancetheoreticallysince1971.Summarizingourresearchresults,severalacademicmonographsandbookchaptershavebeenpublished.Generally,papersaredividedinto8partssuchas(1)MaintenancePolicies,(2)ReliabilityAnalysis,(3)DamageMod-els,(4)ReliabilityApplication,(5)ComputerAnalysis,(6)ManagementModels,(7)FailureDistributions,and(8)OtherStochasticModels.Thischaptermakesmypublicationlistofacademicmonographs,bookchaptersandmainpapersin8partsinorderofdate.Recently,systemshavebecomemuchmorecomplex,andoldplantsandstructureshaveincreasedrapidlyinnearfuture.Inaddition,advancednationshavealmostfinishedpublicinfrastructure.Maintenancepoliciesforindustrialsystemsandpublicinfrastructureshouldbeproperlyandquicklyestablishedaccordingtotheiroccasions.Theseresultsofresearchwouldbeappliedpracticallytoactualreliabilitysystemsbymodifyingandextendingthem.Hereafter,wehavesomeplansinviewsuchasProposalofNewMainte-nancePolicies,AnalysisofMoreRandomSystems,FormulationofSoftwareMaintenanceTheory,ApplicationstoNewManagementModels,andNewSchemesofReliabilityTheory.Suchresearchtargetswouldoffernewin-terestingtheoreticaltopicsforfuturestudies.349 364September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book350ReliabilityModelingwithApplications2PublicationList2.1AcademicMonographs1.NakagawaT.(2005)MaintenanceTheoryofReliability.Springer,London.2.NakagawaT.(2007)ShockandDamageModelsinReliabilityTheory.Springer,London.3.NakagawaT.(2008)AdvancedReliabilityModelsandMaintenancePoli-cies.Springer,London.4.NakamuraS.andNakagawaT.(Eds)(2010)StochasticReliabilityMod-eling,OptimizationandApplications.WorldScientific,Singapore.5.NakagawaT.(2011)StochasticProcesseswithApplicationstoReliabil-ityTheory.Springer,London.6.DohiT.andNakagawaT.(Eds)(2013)StochasticReliabilityandMain-tenanceModeling.Springer,London.7.NakagawaT.(2014)RandomMaintenancePolicies.Springer,London.2.2BookChapters1.NakagawaT.(2000)Imperfectpreventivemaintenancemodels.In:Ben-DayaM.,DuffuaaS.O.andRaoufA.(Eds)Maintenance,ModelingandOptimization.KluwerAcademicPub.,Boston,201–214.2.NakagawaT.(2002)Imperfectpreventivemaintenancemodels.In:Os-akiS.(Ed)StochasticModelsinReliabilityandMaintenance.Springer,Berlin,125–140.3.NakagawaT.(2002)Two-unitredundantmodels.In:OsakiS.(Ed)StochasticModelsinReliabilityandMaintenance.Springer,Berlin,165–191.4.YasuiK.,NakagawaT.andSandohH.(2002)Reliabilitymodelsindatacommunicationsystems.In:OsakiS.(Ed)StochasticModelsinReliabilityandMaintenance.Springer,Berlin,281–306.5.NakagawaT.(2003)Maintenanceandoptimumpolicy.In:PhamH.(Ed)HandbookofReliabilityEngineering.Springer,London,367–395.6.NakagawaT.(2006)Statisticalmodelsonmaintenance.In:PhamH.(Ed)HandbookofEngineeringStatistics.Springer,London,835–848.7.ArafukaM.,NakamuraS.,NakagawaT.andKondoH.(2006)OptimalintervalofCRLissueinPKIarchitecture.In:PhamH.(Ed)Reliabil-ityModeling,AnalysisandOptimization.WorldScientific,Singapore,67–79. 365September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookStudiesonReliabilityandMaintenance3518.SugiuraT.,MizutamiS.andNakagawaT.(2006)Optimalrandomandperiodicinspectionpolicies.In:PhamH.(Ed)ReliabilityModeling,AnalysisandOptimization.WorldScientific,Singapore,393–403.9.MizutaniS.,NakagawaT.andItoK.(2006)Optimalinspectionpoli-ciesforaself-diagnosissystemwithtwotypesofinspections.In:PhamH.(Ed)ReliabilityModeling,AnalysisandOptimization.WorldSci-entific,Singapore,417–428.10.ItoK.andNakagawaT.(2006)Maintenanceofacumulativedamagemodelanditsapplicationtogasturbineengineofco-generationsystem.In:PhamH.(Ed)ReliabilityModeling,AnalysisandOptimization.WorldScientific,Singapore,429–438.11.NakagawaT.andItoK.(2007)Optimalavailabilitymodelsofaphasedarrayradar.In:DohiT.,OsakiS.andSawakiK.(Eds)RecentAd-vancesinStochasticOperationsResearch.WorldScientific,Singapore,115–130.12.NakamuraS.,ArafukaM.andNakagawaT.(2007)OptimalcertificateupdateintervalconsideringcommunicationcostsinPKI.In:DohiT.,OsakiS.andSawakiK.(Eds)RecentAdvancesinStochasticOpera-tionsResearch.WorldScientific,Singapore,235–244.13.NakagawaT.(2008)Replacementandpreventivemaintenancemod-els.In:MisraK.B.(Ed)HandbookofPerformabilityEngineering.Springer,London,807–823.14.NakagawaT.andMizutaniS.(2008)Periodicandsequentialimperfectpreventivemaintenancepoliciesforcumulativedamagemodels.In:PhamH.(Ed)RecentAdvancesinReliabilityandQualityinDesign.Springer,London,85–99.15.MizutaniS.andNakagawaT.(2009)Optimalpolicyforatwo-unitsystemwithtwotypesofinspections.In:DohiT.,OsakiS.andSawakiK.(Eds)RecentAdvancesinStochasticOperationsResearchII.WorldScientific,Singapore,171–181.16.ItoK.andNakagawaT.(2009)Optimalcensoringpoliciesfortheop-erationofadamagesystem.In:DohiT.,OsakiS.andSawakiK.(Eds)RecentAdvancesinStochasticOperationsResearchII.WorldScientific,Singapore,201–210.17.NaruseK.,NakagawaT.andMaejiS.(2009)Optimalsequentialcheck-pointintervalsforerrordetection.In:DohiT.,OsakiS.andSawakiK.(Eds)RecentAdvancesinStochasticOperationsResearchII.WorldScientific,Singapore,213–224.18.NakamuraS.,NakagawaT.andKondoH.(2009)Optimalbackupin- 366October8,201314:34BC:9023-ReliabilityModelingwithApplications2013book352ReliabilityModelingwithApplicationstervalofadatabasesystemusingacontinuousdamagemodel.In:DohiT.,OsakiS.andSawakiK.(Eds)RecentAdvancesinStochasticOperationsResearchII.WorldScientific,Singapore,243–251.19.ZhaoX.andNakagawaT.(2013)Comparisonsofperiodicandrandomreplacementpolicies.In:FrenkelI.,KaragrigoriouA.,LisnianskiA.andKleynerA.V.(Eds)AppliedReliabilityEngineeringandRiskAnal-ysis,ProbabilisticModelsandStatisticalInference,Wiley,NewYork,193–204.2.3Papers(1)MaintenancePolicies1.NakagawaT.andOsakiS.(1974)Optimumreplacementpolicieswithdelay.J.ofAppliedProbability11,102–110.2.NakagawaT.andOsakiS.(1974)Optimumrepairlimitreplacementpolicies.OperationalResearchQuarterly25,311–317.3.OsakiS.andNakagawaT.(1975)Anoteonagereplacement.IEEETr.onReliabilityR-24,92–94.4.NakagawaT.(1976)Onschedulingthedeliveryofspareunits.IEEETr.onReliabilityR-25,35–37,5.NakagawaT.(1977)Optimumpreventivemaintenanceandrepairlimitpoliciesmaximizingtheexpectedearningrate.RAIROOperationsResearch11,103–108.6.MineH.andNakagawaT.(1977)Intervalreliabilityandoptimumpre-ventivemaintenancepolicy.IEEETr.onReliabilityR-26,131–133.7.NakagawaT.andOsakiS.(1977)Discretetimeagereplacementpoli-cies.OperationalResearchQuarterly28,881–885.8.MineH.andNakagawaT.(1978)Agereplacementmodelwithmixedfailuretimes.IEEETr.onReliabilityR-27,173.9.MineH.andNakagawaT.(1978)Asummaryofoptimumpreventivemaintenancepoliciesmaximizingintervalreliability.J.ofOperationsResearchSoc.ofJapan21,205–216.10.NakagawaT.andYasuiK.(1978)ApproximatecalculationofblockreplacementwithWeibullfailuretimes.IEEETr.onReliabilityR-27,268–269.11.NakagawaT.(1979)Replacementproblemofaparallelsysteminran-domenvironment.J.ofAppliedProbability16,203–205.12.NakagawaT.(1979)Thedecisiontoreplaceaunitearlyorlateinanagereplacementproblem.MicroelectronicsandReliability19,265–267. 367September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookStudiesonReliabilityandMaintenance35313.NakagawaT.(1979)Optimumpolicieswhenpreventivemaintenanceisimperfect.IEEETr.onReliabilityR-28,331–332.14.NakagawaT.(1979)Asummaryofblockreplacementpolicies.RAIROOperationsResearch13,351–361.15.NakagawaT.(1979)Furtherresultsofreplacementproblemofaparallelsysteminrandomenvironment.J.ofAppliedProbability16,923–926.16.NakagawaT.(1979)Imperfectpreventive-maintenance.IEEETr.onReliabilityR-28,402.17.NakagawaT.andYasuiK.(1979)Approximatecalculationofinspec-tionpolicywithWeibullfailuretimes.IEEETr.onReliabilityR-28,403–404.18.NakagawaT.(1979)Optimumreplacementpoliciesforausedunit.J.ofOperationsResearchSoc.ofJapan22,338–346.19.NakagawaT.(1980)Optimuminspectionpoliciesforastandbyunit.J.ofOperationsResearchSoc.ofJapan23,13–26.20.NakagawaT.andYasuiK.(1980)Approximatecalculationofoptimalinspectiontimes.J.ofOperationsResearchSoc.ofJapan31,851–853.21.NakagawaT.(1980)Replacementmodelswithinspectionandpreven-tivemaintenance.MicroelectronicsandReliability20,427–433.22.NakagawaT.(1980)Asummaryofimperfectpreventivemaintenancepolicieswithminimalrepair.RAIROOperationsResearch14,249–255.23.NakagawaT.(1980)Meantimetofailurewithpreventivemaintenance.IEEETr.onReliabilityR-29,341.24.NakagawaT.(1980)Replacementpoliciesforaunitwithrandomandwearoutfailures.IEEETr.onReliabilityR-29,342–344.25.NakagawaT.andYasuiK.(1981)Calculationofage-replacementwithWeibullfailuretimes.IEEETr.onReliabilityR-30,163–164.26.NakagawaT.(1981)Modifiedperiodicreplacementwithminimalrepairatfailure.IEEETr.onReliabilityR-30,165–168.27.NakagawaT.(1981)Asummaryofperiodicreplacementwithminimalrepairatfailure.J.ofOperationsResearchSoc.ofJapan24,213–227.28.NakagawaT.(1981)Generalizedmodelsfordeterminingoptimalnum-berofminimalrepairsbeforereplacement.J.ofOperationsResearchSoc.ofJapan24,325–337.29.NakagawaT.andYasuiK.(1982)Boundsofagereplacementtime.MicroelectronicsandReliability22,603–609.30.NakagawaT.(1982)Amodifiedblockreplacementwithtwovariables.IEEETr.onReliabilityR-31,398–400.31.NakagawaT.andKowadaM.(1983)Analysisofasystemwithmin- 368October8,201314:34BC:9023-ReliabilityModelingwithApplications2013book354ReliabilityModelingwithApplicationsimalrepairanditsapplicationtoreplacementpolicy.EuropeanJ.ofOperationsResearch12,176–182.32.NakagawaT.(1983)Optimalnumberoffailuresbeforereplacementtime.IEEETr.onReliabilityR-32,115–116.33.NakagawaT.(1983)Combinedreplacementmodels.RAIROOpera-tionsResearch17,193–203.34.NakagawaT.,NishiK.andSawaY.(1983)Modifiedperiodicpreventivemaintenancepolicies.MicroelectronicsandReliability23,945–951.35.ShimaE.andNakagawaT.(1984)Optimuminspectionpolicyforaprotectivedevice.ReliabilityEngineering7,123–132.36.NakagawaT.(1984)Periodicinspectionpolicywithpreventivemain-tenance.NavalResearchLogisticsQuarterly31,33–40.37.NakagawaT.(1984)Asummaryofdiscretereplacementpolicies.Eu-ropeanJ.ofOperationsResearch17,382–392.38.NakagawaT.(1984)Optimalpolicyofcontinuousanddiscretereplace-mentwithminimalrepairatfailure.NavalResearchLogisticsQuar-terly31,543–550.39.NakagawaT.(1985)Continuousanddiscreteage-replacementpolicies.J.ofOperationsResearchSoc.36,147–154.40.NakagawaT.(1986)Periodicandsequentialpreventivemaintenancepolicies.J.ofAppliedProbability23,536–542.41.NakagawaT.(1986)Modifieddiscretepreventivemaintenancepolicies.NavalResearchLogisticsQuarterly33,703–715.42.NakagawaT.(1987)Modified,discretereplacementmodels.IEEETr.onReliabilityR-36,243–245.43.NakagawaT.andYasuiK.(1987)Optimumpoliciesforasystemwithimperfectmaintenance.IEEETr.onReliabilityR-36,631–633.44.NakagawaT.(1988)Sequentialimperfectpreventivemaintenancepoli-cies.IEEETr.onReliability37,295–298.45.NakagawaT.(1989)AreplacementpolicymaximizingMTTFofasys-temwithspareunits.IEEETr.onReliability38,210–211.46.NakagawaT.(1989)Asummaryofreplacementmodelswithchangingfailuredistributions.RAIROOperationsResearch23,343–353.47.NakagawaT.andYasuiK.(1991)Periodic-replacementmodelswiththresholdlevels.IEEETr.onReliability24,46–49.48.SheuS.H.,KuoC.M.andNakagawaT.(1993)Extendedoptimalagereplacementpolicywithminimalrepair.RAIROOperationsResearch27,337–351.49.NakagawaT.andMurthyD.N.P.(1993)Optimalreplacementpolicies 369September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookStudiesonReliabilityandMaintenance355foratwo-unitsystemwithfailureinteractions.RAIROOperationsResearch27,427–438.50.SheuS.H.,GriffithW.SandNakagawaT.(1995)Extendedoptimalreplacementmodelwithrandomminimalrepaircosts.EuropeanJ.ofOperationsResearch85,636–649.51.NakagawaT.,YasuiK.andSandohH.(2004)Noteonoptimalpar-titionproblemsinreliabilitymodels.J.ofQualityinMaintenanceEngineering10,282–287.52.NakagawaT.andMizutaniS.(2009)Asummaryofmaintenancepoli-ciesforafiniteinterval.ReliabilityEngineering&SystemSafety94,89–96.53.NakagawaT.andMizutaniS.(2009)Optimumproblemsinbackwardtimesofreliabilitymodels.IIETransactions41,1–7.54.ChenM.,NakamuraS.andNakagawaT.(2010)Replacementandpre-ventivemaintenancemodelswithrandomworkingtimes.IEICETrans.FundamentalsE93-A,500–507.55.YunW.Y.andNakagawaT.(2010)Replacementandinspectionpoliciesforproductswithrandomlifecycle.ReliabilityEngineering&SystemSafety95,161–165.56.ChenM.,MizutaniS.andNakagawaT.(2010)Randomandagere-placementpolicies.InternationalJ.ofReliability,QualityandSafetyEngineering17,27–39.57.ChenM.,MizutaniS.andNakagawaT.(2010)Optimalbackwardandbackuppoliciesinreliabilitytheory.J.ofOperationsResearchSoc.ofJapan53,101–118.58.NakagawaT.,MizutaniS.andChenM.(2010)Asummaryofperi-odicandrandominspectionpolicies.ReliabilityEngineering&SystemSafety95,906–911.59.NakagawaT.,ZhaoX.andYunW.Y.(2011)Optimalagereplacementandinspectionpolicieswithrandomfailureandreplacementtimes.InternationalJ.ofReliability,QualityandSafetyEngineering18,1–12.60.NakagawaT.andZhaoX.(2012)Optimizationproblemsofaparallelsystemwitharandomnumberofunits.IEEETr.onReliability61,543–548.61.ZhaoX.andNakagawaT.(2012)Optimizationproblemsofreplace-mentfirstorlastinreliabilitytheory.EuropeanJ.ofOperationalResearch223,141–149.62.NakagawaT.andZhaoX.(2013)Comparisonsofreplacementpolicies 370September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book356ReliabilityModelingwithApplicationswithconstantandrandomtimes.J.ofOperationsResearchSoc.ofJapan56,1–14.(2)ReliabilityAnalysis1.OsakiS.andNakagawaT.(1971)Onatwo-unitstandbyredundantsystemwithstandbyfailure.OperationsResearch19,510–523.2.NakagawaT.andGoelA.L.(1973)Anoteonavailabilityforafiniteinterval.IEEETr.onReliabilityR-22,271–272.3.NakagawaT.andOsakiS.(1974)Stochasticbehaviorofatwo-unitstandbyredundantsystem.INFOR12,66–70.4.NakagawaT.(1974)Theexpectednumberofvisitstostatekbeforeatotalsystemfailureofacomplexsystemwithrepairmaintenance.OperationsResearch22,108–116.5.NakagawaT.andOsakiS.(1974)Stochasticbehaviorofatwo-dissimilar-unitstandbyredundantsystemwithrepairmaintenance.Mi-croelectronicsandReliability13,143–148.6.NakagawaT.andOsakiS.(1974)Optimumpreventivemaintenancepoliciesfora2-unitredundantsystem.IEEETr.onReliabilityR-23,86–91.7.NakagawaT.andOsakiS.(1974)Offtimedistributionsinanalternat-ingrenewalprocesswithreliabilityapplications.MicroelectronicsandReliability13,181–184.8.NakagawaT.andOsakiS.(1974)Optimumpreventivemaintenancepoliciesmaximizingthemeantimetothefirstsystemfailureforatwo-unitstandbyredundantsystem.OptimizationTheoryandApplications14,115–129.9.NakagawaT.andOsakiS.(1974)Combiningdriftandcatastrophicfailuremodes.IEEETr.onReliabilityR-23,278–279.10.NakagawaT.andOsakiS.(1975)Onaterminatingrenewalprocesswithreliabilityapplications.IEEETr.onReliabilityR-24,88–90.11.NakagawaT.andOsakiS.(1975)Stochasticbehaviorof2unitstandbyredundantsystemswithimperfectswitchover.IEEETr.onReliabilityR-24,143–146.12.NakagawaT.,GoelA.L.andOsakiS.(1975)Stochasticbehaviorofanintermittentlyusedsystem.RAIROOperationsResearch2,101–112.13.NakagawaT.andOsakiS.(1975)Stochasticbehaviorofatwo-unitprioritystandbyredundantsystemwithrepair.MicroelectronicsandReliability14,309–313. 371September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookStudiesonReliabilityandMaintenance35714.NakagawaT.andOsakiS.(1975)Thebusyperiodofarepairmanforredundantrepairablesystems.RAIROOperationsResearch3,69–73.15.NakagawaT.andOsakiS.(1975)Stochasticbehavioroftwo-unitpar-allelredundantsystemswithrepairmaintenance.MicroelectronicsandReliability14,457–461.16.NakagawaT.andOsakiS.(1975)Applicationsofthesojourn-timeproblemtoreliability.IEEETr.onReliabilityR-24,301–302.17.NakagawaT.andOsakiS.(1975)Stochasticbehaviorofa2-unitpar-allelfuelchargingsystem.IEEETr.onReliabilityR-24,302–304.18.NakagawaT.andOsakiS.(1976)Reliabilityanalysisofaone-unitsystemwithunrepairablespareunitsanditsoptimizationapplications.OperationalResearchQuarterly27,101–110.19.SuzukiY.,NakagawaT.andSawaY.(1976)Reliabilityanalysisofintermittentlyusedsystemswhenfailuresaredetectedonlyduringausageperiod.MicroelectronicsandReliability15,35–38.20.NakagawaT.andOsakiS.(1976)Analysisofarepairablesystemwhichoperatesatdiscretetimes.IEEETr.onReliabilityR-25,110–112.21.NakagawaT.andOsakiS.(1976)Jointdistributionofuptimeanddowntimeforsomerepairablesystems.J.ofOperationsResearchSoc.ofJapan19,209–216.22.OsakiS.andNakagawaT.(1976)Bibliographyforreliabilityandavail-abilityofstochasticsystems.IEEETr.onReliabilityR-25,284–287.23.NakagawaT.andOsakiS.(1976)Asummaryofoptimumpreventivemaintenancepoliciesforatwo-unitstandbyredundantsystem.ZOROperationsResearch20,171–187.24.NakagawaT.andOsakiS.(1976)Markovrenewalprocesseswithsomenon-regenerationpointsandtheirapplicationstoreliabilitytheory.Mi-croelectronicsandReliability15,633–636.25.MineH.andNakagawaT.(1976)Stochasticbehavioroftwo-unitre-dundantsystemswhichoperateatdiscretetimes.MicroelectronicsandReliability15,551–554.26.NakagawaT.(1977)A2-unitrepairableredundantsystemwithswitch-ingfailure.IEEETr.onReliabilityR-26,128–130.27.NakagawaT.andYasuiK.(1977)Approximatecalculationofsystemavailability.IEEETr.onReliabilityR-26,133–134.28.NakagawaT.(1977)Optimumpreventivemaintenancepoliciesforre-pairablesystems.IEEETr.onReliabilityR-26,168–173.29.NakagawaT.(1978)Reliabilityanalysisofstandbyrepairablesys- 372September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book358ReliabilityModelingwithApplicationstemswhenanemergencyoccurs.MicroelectronicsandReliability17,461–464.30.NakagawaT.(1984)Optimumnumberofunitsforaparallelsystem.J.ofAppliedProbability21,431–436.31.NakagawaT.(1985)Optimizationproblemsink-out-of-nsystems.IEEETr.onReliabilityR-34,248–250.32.YasuiK.,NakagawaT.andOsakiS.(1988)Asummaryofoptimumreplacementpoliciesforaparallelredundantsystem.MicroelectronicsandReliability28,635–641.33.TeramotoK.,NakagawaT.andMotooriM.(1990)Optimalinspectionpolicyforaparallelredundantsystem.MicroelectronicsandReliability30,151–155.34.NakagawaT.andQianC.H.(2002)Noteonreliabilityofseries-parallelandparallel-seriessystems.J.ofQualityinMaintenanceEngineering8,274–280.35.NakagawaT.andYasuiK.(2003)Noteonreliabilityofasystemcom-plexity.MathematicalComputingandModelling38,1365–1371.36.NakagawaT.andYasuiK.(2003)Noteonreliabilityofasystemcom-plexity.J.ofQualityinMaintenanceEngineering9,83–91.37.NakagawaT.andYasuiK.(2005)Noteonoptimalredundantpoliciesforreliabilitymodels.J.ofQualityinMaintenanceEngineering11,82–96.(3)DamageModels1.NakagawaT.andOsakiS.(1974)Someaspectsofdamagemodels.MicroelectronicsandReliability13,253–257.2.NakagawaT.(1975)Oncumulativedamagewithannealing.IEEETr.onReliabilityR-24,90–91.3.NakagawaT.(1976)OnacumulativedamagemodelwithNdifferentcomponents.IEEETr.onReliabilityR-25,112–114.4.NakagawaT.(1976)Onareplacementproblemofacumulativedamagemodel.OperationalResearchQuarterly27,895–900.5.NakagawaT.andKijimaM.(1989)Replacementpoliciesforacumu-lativedamagemodelwithminimalrepairatfailure.IEEETr.onReliability38,581–584.6.KijimaM.andNakagawaT.(1992)Replacementpoliciesofashockmodelwithimperfectpreventivemaintenance.EuropeanJ.ofOpera-tionsResearch12,176–182. 373September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookStudiesonReliabilityandMaintenance3597.SatowT.,YasuiK.andNakagawaT.(1996)Optimalgarbagecollec-tionpoliciesforadatabaseinacomputersystem.RAIROOperationsResearch30,359–372.8.SatowT.,YasuiK.andNakagawaT.(1996)Optimalgarbagecollectionpoliciesforadatabasewithrandomthresholdlevel.ElectronicsandCommunicationsinJapan79,31–40.9.SatowT.andNakagawaT.(1997)Threereplacementmodelswithtwokindsofdamage.MicroelectronicsandReliability37,909–913.10.SatowT.andNakagawaT.(1997)Replacementpoliciesforashockmodelwithtwokindsofdamage.LectureNotesinEconomicsandMathematicalSystem445,188–195.11.SatowT.andNakagawaT.(1997)Optimalreplacementpolicyforacumulativedamagemodelwithdeterioratedinspection.InternationalJ.ofReliability,QualityandSafetyEngineering4,387–393.12.QianC.H.,NakamuraS.andNakagawaT.(1999)Cumulativedamagemodelwithtwokindsofshocksanditsapplicationtothebackuppolicy.J.ofOperationalResearchSoc.ofJapan42,501–511.13.SatowT.,TeramotoK.andNakagawaT.(2000)Optimalreplacementpolicyforacumulativedamagemodelwithtimedeterioration.Math-ematicalandComputerModelling31,313–319.14.QianC.H.,NakamuraS.andNakagawaT.(2000)Replacementpoli-ciesforcumulativedamagemodelwithmaintenancecost.ScientiaeMathematicae3,117–126.15.QianC.H.,ItoK.andNakagawaT.(2005)Optimalpreventivemainte-nancepoliciesforashockmodelwithgivendamagelevel.J.ofQualityinMaintenanceEngineering11,216–227.16.ZhaoX.,QianC.H.andNakagawaT.(2009)Studyonpreventivesoft-warerejuvenationpolicyfortwokindsofbugs.J.ofSystemScienceandInformation7,103–110.17.ZhaoX.andNakagawaT.(2010)Optimalreplacementpoliciesfordamagemodelswiththelimitnumberofshocks.InternationalJ.ofReliabilityandQualityPerformance2,13–20.18.ZhaoX.,NakamuraS.andNakagawaT.(2011)Twogenerationalgarbagecollectionmodelswithmajorcollectiontime.IEICETrans-actionsFundamentalsE94-A,1558–1566.19.ZhaoX.,ZhangH.,QianC.H.,NakagawaT.andNakamuraS.(2012)Replacementmodelsforcombiningadditiveindependentdamages.In-ternationalJ.ofPerformabilityEngineering8,91–100.20.ZhaoX.,NakagawaT.andQianC.H.(2012)Optimalimperfectpreven- 374September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book360ReliabilityModelingwithApplicationstivemaintenancepoliciesforausedsystem.InternationalJ.ofSystemsScience43,1632–164121.ZhaoX.,NakamuraS.andNakagawaT.(2012)Optimaltenuringandmajorcollectiontimesforagenerationalgarbagecollector.Asia-PacificJ.ofOperationalResearch29,1240018(17pages).18.NakamuraS.,ZhaoX.andNakagawaT.(2013)Stochasticmodelingofdatabasebackuppolicyforacomputersystem.J.ofSoftwareEngi-neeringandApplications6,53–58.19.ZhaoX.,QianC.H.andNakagawaT.(2013)Optimalpoliciesforcu-mulativedamagemodelswithmaintenancelastandfirst.ReliabilityEngineering&SystemSafety110,50–59.20.ZhaoX.,NakamuraS.andNakagawaT.(2013)Optimalmaintenancepoliciesforcumulativedamagemodelswithrandomworkingtimes.J.ofQualityinMaintenanceEngineering19,25–37(4)ReliabilityApplications1.NakagawaT.(1976)Onschedulingthedeliveryofspareunits.IEEETr.onReliabilityR-25,35–37.2.NakagawaT.andOsakiS.(1978)Optimumorderingpolicieswithleadtimeforanoperatingunit.RAIROOperationsResearch12,383–393.3.ItoK.andNakagawaT.(1992)Optimalinspectionpoliciesforasysteminstorage.Computers&MathematicswithApplications24,87–90.4.ItoK.,NakagawaT.andNishiK.(1995)Extendedoptimalinspec-tionpoliciesforasysteminstorage.MathematicalComputingandModelling22,83–87.5.ItoK.andNakagawaT.(1995)Anoptimalinspectionpolicyforastor-agesystemwiththreetypesofhazardratefunctions.J.ofOperationsResearchSoc.ofJapan38,423–441.6.ItoK.andNakagawaT.(1995)Anoptimalinspectionpolicyforastoragesystemwithhighreliability.MicroelectronicsandReliability35,875–886.7.ItoK.,TeramotoK.andNakagawaT.(1997)Optimaltimesofburn-intestsformultilevelassemblytimes.LectureNotesinEconomicsandMathematicalSystems445,236–245.8.ItoK.andNakagawaT.(2000)Optimalinspectionpoliciesforastoragesystemwithdegradationatperiodictests.MathematicalComputerModelling31,191–195. 375September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookStudiesonReliabilityandMaintenance3619.SandohH.andNakagawaT.(2003)Howmuchshouldwereweigh?J.ofOperationsResearchSoc.54,318–321.10.ItoK.andNakagawaT.(2003)Optimalself-diagnosispolicyforFADECofgasturbinsengines.MathematicalandComputingMod-eling38,1243–1248.11.ItoK.andNakagawaT.(2004)Comparisonofcyclicanddelayedmain-tenancesforaphasedarrayradar.J.ofOperationsResearchSoc.ofJapan47,51–61.12.SandohH.,IgakiN.andNakagawaT.(2004)Inspectionpolicyforascaleconsideringaccidentaldetections.J.ofQualityinMaintenanceEngineering10,148–153.13.NakagawaT.andItoK.(2008)Optimalmaintenancepoliciesforasystemwithmultiechelonrisks.IEEETr.onSystem,Man,andCy-bernetics38,461–469.14.QianC.H.,ChenJ.andNakagawaT.(2009)Comparisonoftwoin-formationstructureswithnoiseanditsapplicationtoBaysdecisionanalysis.QualityTechnology&QuantitativeManagement6,1–10.15.ChenM.andNakagawaT.(2012)Optimalschedulingofrandomworkswithreliabilityapplications.Asia-PacificJ.ofOperationalResearch29,1250027(14pages).16.ChenM.andNakagawaT.(2013)Optimalredundantsystemsforworkswithrandomprocessingtime.ReliabilityEngineering&SystemSafety116,99–104.(5)ComputerAnalysis1.NakagawaT.(1982)Reliabilityanalysisofacomputersystemwithhiddenfailure.PolicyandInformation6,43–49.2.NakagawaT.,NishiK.andYasuiK.(1984)Optimumpreventivemain-tenancepoliciesforacomputersystemwithresrart.IEEETr.onReliabilityR-33,272–276.3.YasuiK.andNakagawaT.(1989)Asimplefault-diagnosissystemwithperiodictesting.IEEETr.onReliability38,571–572.4.NakagawaT.andYasuiK.(1989)Optimaltesting-policiesforinter-mittentfaults.IEEETr.onReliability38,577–580.5.NakagawaT.,MotooriM.andYasuiK.(1990)Optimaltestingpolicyforacomputersystemwithintermittentfaults.ReliabilityEngineeringandSystemSafety27,213–218. 376September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book362ReliabilityModelingwithApplications6.NakagawaT.,YasuiK.andMotootiM.(1990)Optimalfaultmargininacomputersystem.MicroelectronicsandReliability30,1117–1121.7.YasuiK.,NakagawaT.andKoikeS.(1992)Reliabilityconsiderationonerrorcontrolpoliciesforadatacommunicationsystem.Computers&MathematicswithApplications24,51–55.8.HayashiI.,IshiiN.,SandohH.andNakagawaT.(1995)Optimalcharg-ingtimesofabatteryformemorybackup.MathematicalComputingandModelling22,71–75.9.KoikeS.,NakagawaT.andYasuiK.(1995)Optimalblocklengthforbasicmodedatatransmissioncontrolprocedure.MathematicalCom-putingandModelling22,161–171.10.YasuiK.andNakagawaT.(1995)Reliabilityconsiderationofaselective-repeatARQpolicyforadatacommunicationsystem.Mi-croelectronicsandReliability35,41–44.11.SawadaK.,SandohH.andNakagawaT.(1998)AstudyonARQpoli-ciesfordatatransmissionbasedonKullback-Leiblerinformation.In-ternationalJ.ofReliability,QualityandSafetyEngineering5,5–13.12.SandohH.,HirakoshiH.andNakagawaT.(1998)Anewmodifieddiscretepreventivemaintenancepolicyanditsapplicationtoharddiskmanagement.J.ofQualityinMaintenanceEngineering4,284–290.13.YasuiK.,NakagawaT.andImaizumiM.(1998)ReliabilityevaluationsofhybridARQpoliciesforadatacommunicationsystem.InternationalJ.ofReliability,QualityandSafetyEngineering5,15–28.14.ImaizumiM.,YasuiK.andNakagawaT.(1998)Reliabilityanalysisofmicroprocessorsystemswithwatchdogprocessors.J.ofQualityinMaintenanceEngineering4,263–272.15.ImaizumiM.,YasuiK.andNakagawaT.(2000)Anoptimalnumberofmicroprocessorunitswithwatchdogprocessor.MathematicalandComputerModeling31,183–190.16.QianC.H.,PanY.andNakagawaT.(2002)Optimalpoliciesforadatabasesystemwithtwobackupschemes.RAIROOperationsResearch36,227–235.17.ImaizumiM.,YasuiK.andNakagawaT.(2003)Optimalresetnum-berofmicroprocessorsystemwithnetworkprocessing.ComputerandMathematicswithApplications46,1047–1054.18.ImaizumiM.,YasuiK.andNakagawaT.(2003)Reliabilityofajobexe-cutionprocessusingsignature.MathematicalandComputerModelling38,1219–1223.19.KimuraM.,YasuiK.,NakagawaT.andIshiiN.(2003)Optimalcheck- 377September17,201314:50BC:9023-ReliabilityModelingwithApplications2013bookStudiesonReliabilityandMaintenance363pointintervalofacpmmunicationsystemwithrollbackrecovery.Math-ematicalandComputerModelling38,1303–1311.20.NakamuraS.,QianC.H.,FukumotoS.andNakagawaT.(2003)Op-timalbackuppolicyforadatabasesystemwithincrementalandfullbackups.MathematicalandComputerModelling38,1373–1379.21.KimuraM.,YasuiK.,NakagawaT.andIshiiN.(2006)Reliabilitycon-siderationofamobilecommunicationsystemwithnetworkcongestion.ComputerandMathematicswithApplications51,377–386.22.NakagawaT.,NaruseK.andMaejiS.(2009)RandomcheckpointmodelswithNtandemtasks.IEICETrans.FundamentalsE92-A,1572–1577.23.KimuraM.,ImaizumiM.andNakagawaT.(2013)Reliabilityanalysisofareplicationwithlimitednumberofjournalingfiles.ReliabilityEngineeringandSystemSafety116,105–108.(6)ManagementModels1.NakamuraS.,NakagawaT.andSandohH.(1998)OptimalnumberofsparecashboxesforunmannedbankATMS.RAIROOperationsRe-search32,389–398.2.NakamuraS.,QianC.H.,SandohH.andNakagawaT.(2002)Determi-nationofloaninterestrateconsideringbankruptcyandmortgagecol-lectioncosts.InternationalTr.inOperationsResearch9,695–701.3.NakamuraS.,QianC.H.andNakagawaT.(2003)Anoptimalmainte-nancetimeofautomaticmonitoringsystemofATMwithtwokindsofbreakdowns.ComputerandMathematicswithApplications46,1095–1101.(7)FailureDistributions1.NakagawaT.andOsakiS.(1975)Thediscretedistribution.IEEETr.onReliabilityR-24,300–301.2.NakagawaT.andYodaH.(1977)Relationshipsamongdistributions.IEEETr.onReliabilityR-26,352–353.3.NakagawaT.(1978)Someinequalitiesforfailuredistributions.IEEETr.onReliabilityR-27,58–59.4.NakagawaT.(1978)Discreteextremedistributions.IEEETr.onReli-abilityR-27,367–368.5.NakagawaT.(1984)Discretereplacementmodels.LectureNotesinEconomicsandMathematicalSystems235,38–52. 378September17,201314:50BC:9023-ReliabilityModelingwithApplications2013book364ReliabilityModelingwithApplications(8)OtherStochasticModels1.NakagawaT.andOsakiS.(1974)Amodelforinteractionoftworenewalprocesseswiththresholdlevel.InformationandControl24,1–10.2.NakagawaT.andOsakiS.(1974)Anoteondelaysinducedbyrandomexogenousevents.TransportionScience8,190–192.3.NakagawaT.andOsakiS.(1975)Optimaldentalscheduling.Mathe-maticalBiosciences25,91–104.
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